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Journal articles on the topic 'Penalty parameter and scaling parameter'

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Published: 3 June 2025

Last updated: 6 July 2025

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1

Gao, Xiangyu, Xian Zhang, and Yantao Wang. "A Simple Exact Penalty Function Method for Optimal Control Problem with Continuous Inequality Constraints." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/752854.

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We consider an optimal control problem subject to the terminal state equality constraint and continuous inequality constraints on the control and the state. By using the control parametrization method used in conjunction with a time scaling transform, the constrained optimal control problem is approximated by an optimal parameter selection problem with the terminal state equality constraint and continuous inequality constraints on the control and the state. On this basis, a simple exact penalty function method is used to transform the constrained optimal parameter selection problem into a sequence of approximate unconstrained optimal control problems. It is shown that, if the penalty parameter is sufficiently large, the locally optimal solutions of these approximate unconstrained optimal control problems converge to the solution of the original optimal control problem. Finally, numerical simulations on two examples demonstrate the effectiveness of the proposed method.

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HOTHAZIE, Mihai-Vladut, Georgiana ICHIM, and Mihai-Victor PRICOP. "Development and validation of constraints handling in a Differential Evolution optimizer." INCAS BULLETIN 12, no. 1 (2020): 59–66. http://dx.doi.org/10.13111/2066-8201.2020.12.1.6.

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Research work requires independent, portable optimization tools for many applications, most often for problems where derivability of objective functions is not satisfied. Differential evolution optimization represents an alternative to the more complex, encryption based genetic algorithms. Various packages are available as freeware, but they lack constraints handling, while constrained optimizations packages are commercially available. However, the literature devoted to constraints treatment is significant and the current work is devoted to the implementation of such an optimizer, to be applied in low-fidelity optimization processes. The parameter free penalty scheme is adopted for implementation, and the code is validated against the CEC2006 benchmark test problems and compared with the genetic algorithm in MATLAB. Our paper underlines the implementation of constrained differential evolution by varying two parameters, a predefined parameter for feasibility and the scaling factor, to ensure the convergence of the solution.

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Dong, Zhaonan, and Alexandre Ern. "Hybrid high-order method for singularly perturbed fourth-order problems on curved domains." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 6 (2021): 3091–114. http://dx.doi.org/10.1051/m2an/2021081.

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We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilization operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.

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Qin, Shaopeng, Gaofeng Wei, Zheng Liu, and Xuehui Shen. "Elastodynamic Analysis of Functionally Graded Beams and Plates Based on Meshless RKPM." International Journal of Applied Mechanics 13, no. 04 (2021): 2150043. http://dx.doi.org/10.1142/s1758825121500435.

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In this paper, the reproducing kernel particle method (RKPM) is innovatively extended to the elastodynamic analysis of functionally graded material (FGM). The elastodynamics governing equations of FGM are solved by using the RKPM. The penalty factor method is used to solve the displacement boundary conditions, and the Newmark-[Formula: see text] method is used to discretize the time. The influence of the penalty factor and the scaling parameter is discussed, and the stability and convergence of the RKPM are analyzed. Finally, the correctness of meshless RKPM to solve the elastodynamics of FGM is verified by numerical examples of the functionally graded beams and plates.

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Rajput, Abhishek, Alessandro Roggero, and Nathan Wiebe. "Hybridized Methods for Quantum Simulation in the Interaction Picture." Quantum 6 (August 17, 2022): 780. http://dx.doi.org/10.22331/q-2022-08-17-780.

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Conventional methods of quantum simulation involve trade-offs that limit their applicability to specific contexts where their use is optimal. In particular, the interaction picture simulation has been found to provide substantial asymptotic advantages for some Hamiltonians, but incurs prohibitive constant factors and is incompatible with methods like qubitization. We provide a framework that allows different simulation methods to be hybridized and thereby improve performance for interaction picture simulations over known algorithms. These approaches show asymptotic improvements over the individual methods that comprise them and further make interaction picture simulation methods practical in the near term. Physical applications of these hybridized methods yield a gate complexity scaling as log2&#x2061;&#x039B; in the electric cutoff &#x039B; for the Schwinger Model and independent of the electron density for collective neutrino oscillations, outperforming the scaling for all current algorithms with these parameters. For the general problem of Hamiltonian simulation subject to dynamical constraints, these methods yield a query complexity independent of the penalty parameter &#x03BB; used to impose an energy cost on time-evolution into an unphysical subspace.

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Liu, Zheng, Gaofeng Wei, Zhiming Wang, and Jinwei Qiao. "The Meshfree Analysis of Geometrically Nonlinear Problem Based on Radial Basis Reproducing Kernel Particle Method." International Journal of Applied Mechanics 12, no. 04 (2020): 2050044. http://dx.doi.org/10.1142/s1758825120500441.

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Based on the reproducing kernel particle method (RKPM) and the radial basis function (RBF), the radial basis reproducing kernel particle method (RRKPM) is presented for solving geometrically nonlinear problems. The advantages of the presented method are that it can eliminate the negative effect of diverse kernel functions on the computational accuracy and has greater computational accuracy and better convergence than the RKPM. Using the weak form of Galerkin integration and the Total Lagrangian (T.L.) formulation, the correlation formulae of the RRKPM for geometrically nonlinear problem are obtained. Newton–Raphson (N-R) iterative method is utilized in the process of numerical solution. Moreover, penalty factor, the scaling parameter, the shaped parameter of the RBF and loading step number are discussed. To prove validity of the proposed method, several numerical examples are simulated and compared to finite element method (FEM) solutions.

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Pająk, G. "Planning of Collision-Free Trajectory for Mobile Manipulators." International Journal of Applied Mechanics and Engineering 18, no. 2 (2013): 475–89. http://dx.doi.org/10.2478/ijame-2013-0028.

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A method of planning sub-optimal trajectory for a mobile manipulator working in the environment including obstacles is presented. The path of the end-effector is defined as a curve that can be parameterized by any scaling parameter, the reference trajectory of a mobile platform is not needed. Constraints connected with the existence of mechanical limits for a given manipulator configuration, collision avoidance conditions and control constraints are considered. The motion of the mobile manipulator is planned in order to maximize the manipulability measure, thus to avoid manipulator singularities. The method is based on a penalty function approach and a redundancy resolution at the acceleration level. A computer example involving a mobile manipulator consisting of a nonholonomic platform and a SCARA type holonomic manipulator operating in a two-dimensional task space is also presented.

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8

Talman, Richard. "Scaling behavior of circular colliders dominated by synchrotron radiation." International Journal of Modern Physics A 30, no. 23 (2015): 1544003. http://dx.doi.org/10.1142/s0217751x15440030.

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The scaling formulas in this paper — many of which involve approximation — apply primarily to electron colliders like CEPC or FCC-ee. The more abstract “radiation dominated” phrase in the title is intended to encourage use of the formulas — though admittedly less precisely — to proton colliders like SPPC, for which synchrotron radiation begins to dominate the design in spite of the large proton mass. Optimizing a facility having an electron–positron Higgs factory, followed decades later by a p, p collider in the same tunnel, is a formidable task. The CEPC design study constitutes an initial “constrained parameter” collider design. Here the constrained parameters include tunnel circumference, cell lengths, phase advance per cell, etc. This approach is valuable, if the constrained parameters are self-consistent and close to optimal. Jumping directly to detailed design makes it possible to develop reliable, objective cost estimates on a rapid time scale. A scaling law formulation is intended to contribute to a “ground-up” stage in the design of future circular colliders. In this more abstract approach, scaling formulas can be used to investigate ways in which the design can be better optimized. Equally important, by solving the lattice matching equations in closed form, as contrasted with running computer programs such as MAD, one can obtain better intuition concerning the fundamental parametric dependencies. The ground-up approach is made especially appropriate by the seemingly impossible task of simultaneous optimization of tunnel circumference for both electrons and protons. The fact that both colliders will be radiation dominated actually simplifies the simultaneous optimization task. All GeV scale electron accelerators are “synchrotron radiation dominated”, meaning that all beam distributions evolve within a fraction of a second to an equilibrium state in which “heating” due to radiation fluctuations is canceled by the “cooling” in RF cavities that restore the lost energy. To the contrary, until now, the large proton to electron mass ratio has caused synchrotron radiation to be negligible in proton accelerators. The LHC beam energy has still been low enough that synchrotron radiation has little effect on beam dynamics; but the thermodynamic penalty in cooling the superconducting magnets has still made it essential for the radiated power not to be dissipated at liquid helium temperatures. Achieving this has been a significant challenge. For the next generation p, p collider this will be even more true. Furthermore, the radiation will effect beam distributions on time scales measured in minutes, for example causing the beams to be flattened, wider than they are high. In this regime scaling relations previously valid only for electrons will be applicable also to protons.

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Weisenthal, Samuel, Samuel J. Weisenthal, Caroline Quill, et al. "2416." Journal of Clinical and Translational Science 1, S1 (2017): 17–18. http://dx.doi.org/10.1017/cts.2017.75.

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OBJECTIVES/SPECIFIC AIMS: Our objective was to develop and evaluate a machine learning pipeline that uses electronic health record (EHR) data to predict acute kidney injury (AKI) during rehospitalization for patients who did not have an AKI episode in their most recent hospitalization. METHODS/STUDY POPULATION: The protocol under which this study falls was given exempt status by our institutional review board. The fully deidentified data set, containing all adult hospital admissions during a 2-year period, is a combination of administrative, laboratory, and pharmaceutical information. The administrative data set includes International Classification of Diseases, 9th Revision (ICD-9) diagnosis and procedure codes, Current Procedural Terminology, 4th Edition (CPT-4) procedure codes, diagnosis-related grouping (DRG) codes, locations visited in the hospital, discharge disposition, insurance, marital status, gender, age, ethnicity, and total length of stay. The laboratory data set includes bicarbonate, chloride, calcium, anion gap, phosphate, glomerular filtration rate, creatinine, urea nitrogen, albumin, total protein, liver function enzymes, and hemoglobin A1c. The pharmacy data set includes, for each medication, a description, pharmacologic class and subclass, and therapeutic class. Data preprocessing was performed using Python library Pandas (McKinney, 2011). Top-level binary representation (Singh, 2015) was used for diagnosis and procedure codes. Categorical variables were transformed via 1-hot encoding. Previous admissions were collapsed using rules informed by domain expertise (eg, the most recent age or sum of assigned diagnosis codes were retained as elements in the feature vector). We excluded any patient without at least 1 rehospitalization during the time window. We excluded any admission with or without AKI where AKI was also present in the most recent hospitalization. For comparison, we do not exclude such admissions for an identical experiment in which we considered any AKI event as a positive sample (regardless of AKI presence in the most recent hospitalization). We defined an AKI event as an assignment of any of the acute kidney failure (AKF) ICD-9 codes [584.5, AKF with lesion of tubular necrosis, 584.6, AKF with lesion of renal cortical necrosis, 584.7, AKF with lesion of renal medullary (papillary) necrosis, 584.8, AKF with other specified pathological lesion in kidney, or 584.9, AKF, unspecified]. Since diagnosis codes are believed to be specific but not sensitive for AKI (Waikar, 2006), we supplemented them using creatinine for patients who had laboratory values. Diagnosis was made according to the Kidney Disease: Improving Global Outcomes (KDIGO) Practice Guidelines (AKI defined as a 1.5-fold or greater increase in serum creatinine from baseline within 7 d or 0.3 mg/dL or greater increase in serum creatinine within 48 h). We report preliminary model discrimination via area under the receiver operating characteristic curve (AUC) using k-fold cross validation grouped by patient identifier (to ensure that admissions from the same patient would not appear in the training and validation set). It was confirmed that the prevalence of positive samples in the entire data set was maintained in each fold. Python library Sci-kit Learn (Pedregosa, 2011) was used for pipeline development, which consisted of imputation, scaling, and hyper-parameter tuning for penalized (l1 and l2 norm) logistic regression, random forest, and multilayer perceptron classifiers. All experiments were stored in IPython (Pérez, 2007) notebooks for easy viewing and result reproduction. RESULTS/ANTICIPATED RESULTS: There were 107,036 adult patients that accounted for 199,545 admissions during a 2-year window. Per admission, there were at most 54 ICD-9 diagnoses, 38 ICD-9 procedures, 314 CPT-4 procedures, and 25 hospital locations visited. The admissions were 55% female, the average age was 46±standard deviation 20, and average length of stay was 2.5±8.0 days. We excluded 2360 admissions that involved an AKI event that directly followed an admission with an AKI event and 4130 admissions that did not involve an AKI event but directly followed an admission with an AKI event. In total, there were 4561 (5.3%) positive samples (AKI during rehospitalization without AKI in the previous stay) generated by 3699 unique patients and 81,458 negative samples (non-AKI during rehospitalization without AKI in the previous stay) generated by 31,831 unique patients. When using any AKI event as a positive sample (regardless of whether or not AKI was in the most recent stay), the prevalence was 7.3% (6921 positive samples generated by 4395 unique patients and 85,588 negative samples generated by 33,287 unique patients). Best results were achieved with a code precision of 3 digits for which we had a total of 4556 features per patient. Fitted hyper-parameters corresponding to each classifier were logistic regression with l1 penalty C as 2×10−3; logistic regression with l2 penalty C as 1×10−6; random forest number of estimators as 100, maximum depth as 3, minimum samples per leaf as 50, minimum samples per split as 10, and entropy as the splitting criterion; and multilayer perceptron l2 regularization parameter α as 15, architecture as 1 hidden layer with 5 units, and learning rate as 0.001. Five-fold stratified cross validation on the development set yielded AUC for logistic regression with l1 penalty average 0.830±0.006, logistic regression with l2 penalty 0.796±0.007, random forest 0.828±0.007, and multilayer perceptron 0.841±0.005. In an identical experiment for which an AKI event was considered a positive sample regardless of AKI presence in the most recent stay, we had 4592 features per sample with the same code precision. Five-fold stratified cross validation on the development set with identical settings for the hyper-parameters yielded AUC for logistic regression with l1 penalty average 0.850±0.004, logistic regression with l2 penalty 0.819±0.006, random forest 0.853±0.004, and multilayer perceptron 0.853±0.006. DISCUSSION/SIGNIFICANCE OF IMPACT: Our objective was to investigate the feasibility of using machine learning methods on EHR data to provide a personalized risk assessment for “unexpected” AKI in rehospitalized patients. Preliminary model discrimination was good, suggesting that this approach is feasible. Such a model could aid clinicians to recognize AKI risk in unsuspicious patients. The authors recognize several limitations. Since our data set corresponds to a time-window sample, patients with high frequency of hospital utilization are likely overrepresented. Similarly, our data set contains records from only 1 hospital network. Although we supplement with laboratory-based diagnosis, using diagnosis codes as labels is problematic as numerous reports suggest low sensitivity of codes for AKI. Future work includes calibration analysis, incremental updating (“online learning”), and a representation learning-based (“deep learning”) extension of the model.

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Ji, Xia, Jiguang Sun, and Yang Yang. "Optimal penalty parameter forC0IPDG." Applied Mathematics Letters 37 (November 2014): 112–17. http://dx.doi.org/10.1016/j.aml.2014.06.001.

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Aliaga-Aguilar, Hugo, and Cristina Cuerno-Rejado. "Generic parameter penalty architecture." Structural and Multidisciplinary Optimization 58, no. 4 (2018): 1559–69. http://dx.doi.org/10.1007/s00158-018-1979-2.

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12

Rubinov, A. M., X. Q. Yang, and A. M. Bagirov. "Penalty functions with a small penalty parameter." Optimization Methods and Software 17, no. 5 (2002): 931–64. http://dx.doi.org/10.1080/1055678021000066058.

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13

Straßer, Sebastian, and Hans-Georg Herzog. "Analysis of penalty parameters for interior penalty Galerkin methods." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 38, no. 5 (2019): 1401–12. http://dx.doi.org/10.1108/compel-12-2018-0514.

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Purpose The purpose of this paper is to analyse the influence of penalty parameters for an interior penalty Galerkin method, namely, the symmetric interior penalty Galerkin method. Design/methodology/approach First of all, the solution of a simple model problem is computed and compared to the exact solution, which is a periodic function. Afterwards, a two-dimensional magnetostatic field problem described by the magnetic vector potential A is considered. In particular, penalty parameters depending on the polynomial degree, the properties of the elements and the material are considered. The analysis is performed by varying the polynomial degree and the mesh sizes on a structured and an unstructured mesh. Additionally, the penalty parameter is varied in a specific range. Findings Choosing the penalty parameter correctly plays an important role as the stability and the convergence of the numerical scheme can be affected. For a structured mesh, a limiting value for the penalty parameter can be calculated beforehand, whereas for an unstructured mesh, the choice of the penalty parameter can be cumbersome. Originality/value This paper shows that there exist different penalty parameters which can be taken into account to solve the considered problems. One can choose a global penalty parameter to obtain a stable solution, which is a sharp estimation. There has always to be the consideration to guarantee the coercivity of the bilinear form while minimising the number of iterations.

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Messac, Achille, Michael P. Martinez, and Timothy W. Simpson. "Introduction of a Product Family Penalty Function Using Physical Programming." Journal of Mechanical Design 124, no. 2 (2002): 164–72. http://dx.doi.org/10.1115/1.1467602.

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In an effort to increase customization for today’s highly competitive global markets, many companies are looking to product families to increase product variety and shorten product lead-times while reducing costs. The key to a successful product family is the common product platform around which the product family is derived. Building on our previous work in product family design, we introduce a product family penalty function (PFPF) in this paper to aid in the selection of common and scaling parameters for families of products derived from scalable product platforms. The implementation of the PFPF utilizes the powerful physical programming paradigm to formulate the problem in terms of physically meaningful parameters. To demonstrate the proposed approach, a family of electric motors is developed and compared against previous results. We find that the PFPF enables us to properly balance commonality and performance within the product family through the judicious selection of the common parameters that constitute the product platform and the scaling parameters used to instantiate the product family.

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Clevenhaus, Anna, Matthias Ehrhardt, Michael Günther, and Daniel Ševčovič. "Pricing American Options with a Non-Constant Penalty Parameter." Journal of Risk and Financial Management 13, no. 6 (2020): 124. http://dx.doi.org/10.3390/jrfm13060124.

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As the American early exercise results in a free boundary problem, in this article we add a penalty term to obtain a partial differential equation, and we also focus on an improved definition of the penalty term for American options. We replace the constant penalty parameter with a time-dependent function. The novelty and advantage of our approach consists in introducing a bounded, time-dependent penalty function, enabling us to construct an efficient, stable, and adaptive numerical approximation scheme, while in contrast, the existing standard approach to the penalisation of the American put option-free boundary problem involves a constant penalty parameter. To gain insight into the accuracy of our proposed extension, we compare the solution of the extension to standard reference solutions from the literature. This illustrates the improvement of using a penalty function instead of a penalising constant.

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Lewis, Thomas, Aaron Rapp, and Yi Zhang. "Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs." Electronic Journal of Differential Equations, Conference 26 (August 25, 2022): 123–38. http://dx.doi.org/10.58997/ejde.conf.26.l1.

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This article analyzes the effect of the penalty parameter used in symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE). DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error. For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html

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Shandiz, Roohollah Aliakbari, and Emran Tohidi. "Decrease of the Penalty Parameter in Differentiable Penalty Function Methods." Theoretical Economics Letters 01, no. 01 (2011): 8–14. http://dx.doi.org/10.4236/tel.2011.11003.

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Janesch, S. M. H. "Exact penalty functions a lower bound to the penalty parameter." International Mathematical Forum 2 (2007): 75–86. http://dx.doi.org/10.12988/imf.2007.07005.

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Singh, I. V. "A Numerical Study of Weight Functions, Scaling, and Penalty Parameters for Heat Transfer Applications." Numerical Heat Transfer, Part A: Applications 47, no. 10 (2005): 1025–53. http://dx.doi.org/10.1080/10407780590926183.

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Zheng, Jie, Jiaxu Ning, Hongfeng Ma, and Ziyi Liu. "A Dynamic Parameter Tuning Strategy for Decomposition-Based Multi-Objective Evolutionary Algorithms." Applied Sciences 14, no. 8 (2024): 3481. http://dx.doi.org/10.3390/app14083481.

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The penalty-based boundary cross-aggregation (PBI) method is a common decomposition method of the MOEA/D algorithm, but the strategy of using a fixed penalty parameter in the boundary cross-aggregation function affects the convergence of the populations to a certain extent and is not conducive to the maintenance of the diversity of boundary solutions. To address the above problems, this paper proposes a penalty boundary crossing strategy (DPA) for MOEA/D to adaptively adjust the penalty parameter. The strategy adjusts the penalty parameter values according to the state of uniform distribution of solutions around the weight vectors in the current iteration period, thus helping the optimization process to balance convergence and diversity. In the experimental part, we tested the MOEA/D-DPA algorithm with several MOEA/D improved algorithms on the classical test set. The results show that the MOEA/D with the DPA has better performance than the MOEA/D with the other decomposition strategies.

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Mongeau, Marcel, and Annick Sartenaer. "Automatic decrease of the penalty parameter in exact penalty function methods." European Journal of Operational Research 83, no. 3 (1995): 686–99. http://dx.doi.org/10.1016/0377-2217(93)e0339-y.

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Pantoja, J. F. A. De O., and D. Q. Mayne. "Exact penalty function algorithm with simple updating of the penalty parameter." Journal of Optimization Theory and Applications 69, no. 3 (1991): 441–67. http://dx.doi.org/10.1007/bf00940684.

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Liu, Lidong, Jifeng Hu, Zishu He, Chunlin Han, Huiyong Li, and Jun Li. "A Velocity Measurement Method Based on Scaling Parameter Estimation of a Chaotic System." Metrology and Measurement Systems 18, no. 2 (2011): 275–82. http://dx.doi.org/10.2478/v10178-011-0009-1.

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A Velocity Measurement Method Based on Scaling Parameter Estimation of a Chaotic SystemIn this paper, we propose a new method of measuring the target velocity by estimating the scaling parameter of a chaos-generating system. First, we derive the relation between the target velocity and the scaling parameter of the chaos-generating system. Then a new method for scaling parameter estimation of the chaotic system is proposed by exploiting the chaotic synchronization property. Finally, numerical simulations show the effectiveness of the proposed method in target velocity measurement.

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Takano, H., and Y. Saito. "Order Parameter and Finite-Size Scaling." Progress of Theoretical Physics 73, no. 6 (1985): 1369–76. http://dx.doi.org/10.1143/ptp.73.1369.

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Lin, Chih-Jen. "Formulations of Support Vector Machines: A Note from an Optimization Point of View." Neural Computation 13, no. 2 (2001): 307–17. http://dx.doi.org/10.1162/089976601300014547.

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In this article, we discuss issues about formulations of support vector machines (SVM) from an optimization point of view. First, SVMs map training data into a higher- (maybe infinite-) dimensional space. Currently primal and dual formulations of SVM are derived in the finite dimensional space and readily extend to the infinite-dimensional space. We rigorously discuss the primal-dual relation in the infinite-dimensional spaces. Second, SVM formulations contain penalty terms, which are different from unconstrained penalty functions in optimization. Traditionally unconstrained penalty functions approximate a constrained problem as the penalty parameter increases. We are interested in similar properties for SVM formulations. For two of the most popular SVM formulations, we show that one enjoys properties of exact penalty functions, but the other is only like traditional penalty functions, which converge when the penalty parameter goes to infinity.

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Stetsyuk, Petro, Andreas Fischer, and Olha Khomiak. "Maximum Penalty Function in Linear Programming." Physico-mathematical modelling and informational technologies, no. 33 (September 6, 2021): 156–60. http://dx.doi.org/10.15407/fmmit2021.33.156.

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A linear program can be equivalently reformulated as an unconstrained nonsmooth minimization problem, whose objective is the sum of the original objective and a penalty function with a sufficiently large penalty parameter. The article presents two methods for choosing this parameter. The first one applies to linear programs with usual linear inequality constraints. Then, we use a corresponding theorem by N.Z. Shor on the equivalence of a convex program to an unconstrained nonsmooth minimization problem. The second method is for linear programs of a special type. This means that all inequalities are of the form that a linear expression on the left-hand side is less or equal to a positive constant on the right-hand side. For this special type, we use a corresponding theorem of B.N. Pshenichny on establishing a penalty parameter for convex programs. For differently sized linear programs of the special type, we demonstrate that suitable penalty parameters can be computed by a procedure in GNU Octave based on GLPK software.

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Gao, Cuixia, Naiyan Wang, Qi Yu, and Zhihua Zhang. "A Feasible Nonconvex Relaxation Approach to Feature Selection." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (2011): 356–61. http://dx.doi.org/10.1609/aaai.v25i1.7921.

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Variable selection problems are typically addressed under apenalized optimization framework. Nonconvex penalties such as the minimax concave plus (MCP) and smoothly clipped absolute deviation(SCAD), have been demonstrated to have the properties of sparsity practically and theoretically. In this paper we propose a new nonconvex penalty that we call exponential-type penalty. The exponential-type penalty is characterized by a positive parameter,which establishes a connection with the ell0 and ell1 penalties.We apply this new penalty to sparse supervised learning problems. To solve to resulting optimization problem, we resort to a reweighted ell1 minimization method. Moreover, we devise an efficient method for the adaptive update of the tuning parameter. Our experimental results are encouraging. They show that the exponential-type penalty is competitive with MCP and SCAD.

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Chen, Yan. "Impact Assessment of Job Machine Factors on Scaling Parameters." Advanced Materials Research 340 (September 2011): 23–29. http://dx.doi.org/10.4028/www.scientific.net/amr.340.23.

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Composite dispatching rules are of great importance in solving realistic scheduling problems because they are fast and easy to implement. The successful implementation of composite dispatching rules depends on their scaling parameter values, while further the appropriate scaling parameter values are determined by specific problem instance features, i.e. job machine factors. In this paper, a Face-centered Cube experimental design is proposed to study the impact of job machine factors on scaling parameter values when Apparent Tardiness Cost with Setups (ATCS) rule is used to schedulePm|sjk|ΣwjTjproblem. The dynamics between the job machine factors and the good scaling parameter values are revealed by experimental results.

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Wang, Yanxin, and Li Zhu. "Variable Selection and Parameter Estimation with the Atan Regularization Method." Journal of Probability and Statistics 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/6495417.

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Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by penalized least squares using various penalty functions. In this paper, an arctangent type penalty which very closely resemblesl0penalty is proposed; we call it Atan penalty. The Atan-penalized least squares procedure is shown to consistently select the correct model and is asymptotically normal, provided the number of variables grows slower than the number of observations. The Atan procedure is efficiently implemented using an iteratively reweighted Lasso algorithm. Simulation results and data example show that the Atan procedure with BIC-type criterion performs very well in a variety of settings.

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Burachik, Regina S., and C. Yalçın Kaya. "An augmented penalty function method with penalty parameter updates for nonconvex optimization." Nonlinear Analysis: Theory, Methods & Applications 75, no. 3 (2012): 1158–67. http://dx.doi.org/10.1016/j.na.2011.03.013.

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Shahbazi, Khosro. "An explicit expression for the penalty parameter of the interior penalty method." Journal of Computational Physics 205, no. 2 (2005): 401–7. http://dx.doi.org/10.1016/j.jcp.2004.11.017.

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32

Bhagat, Amol, Nilesh Kshirsagar, Priti Khodke, Kiran Dongre, and Sadique Ali. "Penalty Parameter Selection for Hierarchical Data Stream Clustering." Procedia Computer Science 79 (2016): 24–31. http://dx.doi.org/10.1016/j.procs.2016.03.005.

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Liu, Yun, Jie Lian, Michael R. Bartolacci, and Qing-An Zeng. "Density-Based Penalty Parameter Optimization on C-SVM." Scientific World Journal 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/851814.

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The support vector machine (SVM) is one of the most widely used approaches for data classification and regression. SVM achieves the largest distance between the positive and negative support vectors, which neglects the remote instances away from the SVM interface. In order to avoid a position change of the SVM interface as the result of an error system outlier, C-SVM was implemented to decrease the influences of the system’s outliers. Traditional C-SVM holds a uniform parameterCfor both positive and negative instances; however, according to the different number proportions and the data distribution, positive and negative instances should be set with different weights for the penalty parameter of the error terms. Therefore, in this paper, we propose density-based penalty parameter optimization of C-SVM. The experiential results indicated that our proposed algorithm has outstanding performance with respect to both precision and recall.

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Benchakroun, Abdelhamid, Jean-Pierre Dussault, and Abdelatif Mansouri. "A two parameter mixed interior-exterior penalty algorithm." ZOR Zeitschrift f�r Operations Research Mathematical Methods of Operations Research 41, no. 1 (1995): 25–55. http://dx.doi.org/10.1007/bf01415063.

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Crisci, Serena, Valentina De Simone, and Marco Viola. "On the Adaptive Penalty Parameter Selection in ADMM." Algorithms 16, no. 6 (2023): 264. http://dx.doi.org/10.3390/a16060264.

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Many data analysis problems can be modeled as a constrained optimization problem characterized by nonsmooth functionals, often because of the presence of ℓ1-regularization terms. One of the most effective ways to solve such problems is through the Alternate Direction Method of Multipliers (ADMM), which has been proved to have good theoretical convergence properties even if the arising subproblems are solved inexactly. Nevertheless, experience shows that the choice of the parameter τ penalizing the constraint violation in the Augmented Lagrangian underlying ADMM affects the method’s performance. To this end, strategies for the adaptive selection of such parameter have been analyzed in the literature and are still of great interest. In this paper, starting from an adaptive spectral strategy recently proposed in the literature, we investigate the use of different strategies based on Barzilai–Borwein-like stepsize rules. We test the effectiveness of the proposed strategies in the solution of real-life consensus logistic regression and portfolio optimization problems.

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Song, Jintao, Wenqi Lu, Yunwen Lei, Yuchao Tang, Zhenkuan Pan, and Jinming Duan. "Optimizing ADMM and Over-Relaxed ADMM Parameters for Linear Quadratic Problems." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 8 (2024): 8117–25. http://dx.doi.org/10.1609/aaai.v38i8.28651.

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The Alternating Direction Method of Multipliers (ADMM) has gained significant attention across a broad spectrum of machine learning applications. Incorporating the over-relaxation technique shows potential for enhancing the convergence rate of ADMM. However, determining optimal algorithmic parameters, including both the associated penalty and relaxation parameters, often relies on empirical approaches tailored to specific problem domains and contextual scenarios. Incorrect parameter selection can significantly hinder ADMM's convergence rate. To address this challenge, in this paper we first propose a general approach to optimize the value of penalty parameter, followed by a novel closed-form formula to compute the optimal relaxation parameter in the context of linear quadratic problems (LQPs). We then experimentally validate our parameter selection methods through random instantiations and diverse imaging applications, encompassing diffeomorphic image registration, image deblurring, and MRI reconstruction.

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Zhao, Xin-Yuan, and Liang Chen. "The Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion." Asia-Pacific Journal of Operational Research 37, no. 04 (2020): 2040001. http://dx.doi.org/10.1142/s0217595920400011.

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In this paper, we conduct a convergence rate analysis of the augmented Lagrangian method with a practical relative error criterion designed in Eckstein and Silva [Mathematical Programming, 141, 319–348 (2013)] for convex nonlinear programming problems. We show that under a mild local error bound condition, this method admits locally a Q-linear rate of convergence. More importantly, we show that the modulus of the convergence rate is inversely proportional to the penalty parameter. That is, an asymptotically superlinear convergence is obtained if the penalty parameter used in the algorithm is increasing to infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if the penalty parameter is fixed but it is sufficiently large. Besides, as a byproduct, the convergence, as well as the convergence rate, of the distance from the primal sequence to the solution set of the problem is obtained.

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Wu, Qiong, and Zijun Hao. "Completely Smooth Lower-Order Penalty Approach for Solving Second-Order Cone Mixed Complementarity Problems." Mathematics 13, no. 5 (2025): 690. https://doi.org/10.3390/math13050690.

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In this paper, a completely smooth lower-order penalty method for solving a second-order cone mixed complementarity problem (SOCMCP) is studied. Four distinct types of smoothing functions are taken into account. According to this method, SOCMCP is approximated by asymptotically completely smooth lower-order penalty equations (CSLOPEs), which includes penalty and smoothing parameters. Under mild assumptions, the main results show that as the penalty parameter approaches positive infinity and the smooth parameter monotonically decreases to zero, the solution sequence of asymptotic CSLOPEs converges exponentially to the solution of SOCMCP. An algorithm based on this approach is developed, and numerical experiments demonstrate its feasibility. The performance profile of four specific smooth functions is given. The final results show that the numerical performance of CSLOPEs is better than that of a smooth-like lower-order penalty method.

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Mozolevski, I., and P. R. Bösing. "Sharp Expressions for the Stabilization Parameters in Symmetric Interior-penalty Discontinuous Galerkin Finite Element Approximations of Fourth-order Elliptic Problems." Computational Methods in Applied Mathematics 7, no. 4 (2007): 365–75. http://dx.doi.org/10.2478/cmam-2007-0022.

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Abstract In this paper, we derive explicit expressions for the penalty parameters appearing in symmetric and semi-symmetric interior-penalty discontinuous Galerkin finite element method (DGFEM) for fourth-order elliptic problems. We demonstrate the sharpness of the theoretically predicted penalty parameter values through numerical experiments.

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Askes, Harm, and Sinniah Ilanko. "The use of negative penalty functions in linear systems of equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2074 (2006): 2965–75. http://dx.doi.org/10.1098/rspa.2006.1716.

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Contrary to what is commonly thought, it is possible to obtain convergent results with negative (rather than positive) penalty functions. This has been shown and proven on various occasions for vibration analysis, but in this contribution it will also be shown and proven for systems of linear equations subjected to one or more constraints. As a key ingredient in the developed arguments, a pseudo-force is identified as the derivative of the constrained degree of freedom with respect to the inverse of the penalty parameter. Since this pseudo-force can be proven to be constant for large absolute values of the penalty parameter, it follows that the exact solution is bounded by the results obtained with negative and positive penalty parameters. The mathematical proofs are presented and two examples are shown to illustrate the principles.

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Lubis, Fadil Ansori, Rahmat Subarkah, and Arifia Ekayuliana. "Pengembangan Simulasi Pada Parameter Suhu Jalur Injeksi Brine di Production Optimizaton Unit (POU) PT.XYZ." Jurnal Mekanik Terapan 3, no. 3 (2022): 79–85. http://dx.doi.org/10.32722/jmt.v3i3.4819.

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Scaling silika merupakan masalah utama dalam proses pemanfaatan energi panas bumi. Scaling silika paling banyak terjadi pada PLTP dengan karakteristik reservoir yang memproduksi fluida 2-fasa. Pilot Testing diperlukan untuk mengetahui parameter dan teknologi yang optimal dalam memitigasi scaling silika. Production Optimization Unit (POU) merupakan unit untuk melakukan pilot testing. Penentuan parameter operasi POU menggunakan simulasi (test planer) secara general dengan algoritma yang terkunci dan tingkat keakuratannya masih rendah. Penelitian ini bertujuan untuk melakukan pengembangan simulasi pada parameter suhu jalur injeksi brine di POU dengan menggunakan prangkat lunak EES dan melakukan verifikasi hasil simulasi dengan hasil pengoperasian POU secara eksperimental. Metode penelitian yang digunakan pada penelitian ini adalah kuantitatif komparatif, atau membandingkan secara langsung data hasil simulasi dan eksperimen untuk memverifikasi. Berdasarkan perbandingan data verifikasi simulasi, pengembangan simulasi berhasil meningkatkan akurasi sebesar 6.471%. Algoritma yang dibuat bersifat fleksibel sehingga engineer PT.XYZ dapat leluasa melakukan rekayasa algoritma dan parameter. Rekayasa diperlukan guna mendapatkan parameter-parameter yang optimum sebagai dasar pengoperasian POU.

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Filip, Ioan, Florin Dragan, Iosif Szeidert, and Adriana Albu. "Minimum-Variance Control System with Variable Control Penalty Factor." Applied Sciences 10, no. 7 (2020): 2274. http://dx.doi.org/10.3390/app10072274.

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The present paper proposes (as the main contribution) an additional self-tuning mechanism for an adaptive minimum-variance control system, whose main goal is to extend its functionality for a large value range of unmeasurable perturbations which disturb the controlled process. Through the standard design procedure, a minimum variance controller uses by default an internal self-tuning mechanism based on the process parameter estimates. However, the main parameter which overwhelmingly influences the control performance is the control penalty factor ( ρ ) . This parameter weights the term that describes the control variance in a criterion function whose minimization is the starting point of the control law design. The classical minimum-variance control involves an off-line tuning of this parameter, its value being set as constant throughout the entire operating regime. Based on the measurement of the process output error, the contribution of the proposed strategy consists in a real-time tuning of the control penalty factor, to ensure the stability of the control system, even under conditions of high disturbances. The proposed tuning mechanism adjusts this parameter by implementing a bipositional switching strategy based on a sharp hysteresis loop. Therefore, instead of the standard solution that involves a constant value of the control penalty factor ρ (a priori computed and set), this paper proposes a dual value for this controller parameter. The main objective is to allow the controlled process to operate in a stable fashion even in more strongly disturbed regimes (regimes where the control system becomes unstable and is usually switched off for safety reasons). To validate the proposed strategy, an induction generator integrated into a wind energy conversion system was considered as controlled plant. Operating under the action of strong disturbances (wind gusts, electrical load variations), the extension of safe operating range (thus avoiding the system disengagement) is an important goal of such a control system.

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Xu, Puer, Niansheng Cheng, and Maoxing Wei. "Length Parameter for Scaling Abutment Scour Depth." Water 12, no. 12 (2020): 3508. http://dx.doi.org/10.3390/w12123508.

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Flow constriction caused by bridge abutment increases bed shear stress and thus enhances local scour. For scaling the maximum scour depth at the abutment, either abutment length or flow depth has been empirically used in previous studies. By performing a step-by-step analysis, this study proposes a new length scale, which is able to represent combined effects of abutment length, approach flow depth and channel width. Physically, the new length scale describes the maximum possible dimension of the associated vortex system (or large-scale turbulence). Six series of data compiled from the published literature were used in the analysis. The results indicate that the new length scale helps improve the agreement of predictions with the experimental data.

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Feingold, Mario, Alessandro Gioletta, Felix M. Izrailev, and Luca Molinari. "Two-parameter scaling in the Wigner ensemble." Physical Review Letters 70, no. 19 (1993): 2936–39. http://dx.doi.org/10.1103/physrevlett.70.2936.

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Lim, C. Y., M. M. Meerschaert, and H. P. Scheffler. "Parameter estimation for operator scaling random fields." Journal of Multivariate Analysis 123 (January 2014): 172–83. http://dx.doi.org/10.1016/j.jmva.2013.09.010.

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Yang, Kyung-won, and Tai-yong Lee. "Heuristic scaling method for efficient parameter estimation." Chemical Engineering Research and Design 88, no. 5-6 (2010): 520–28. http://dx.doi.org/10.1016/j.cherd.2009.09.017.

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Kuznetsov, A. P., S. P. Kuznetsov, and I. R. Sataev. "Three-parameter scaling for one-dimensional maps." Physics Letters A 189, no. 5 (1994): 367–73. http://dx.doi.org/10.1016/0375-9601(94)90018-3.

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Kehlenbeck, Ralf, John Yates, Renzo Di Felice, Hermann Hofbauer, and Reinhard Rauch. "Novel scaling parameter for circulating fluidized beds." AIChE Journal 47, no. 3 (2001): 582–89. http://dx.doi.org/10.1002/aic.690470308.

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Jalnine, Aleksej, and Sergey Kuznetsov. "Universality and scaling in the circle map under external periodic driving." Izvestiya VUZ. Applied Nonlinear Dynamics 10, no. 6 (2003): 3–15. http://dx.doi.org/10.18500/0869-6632-2002-10-6-3-15.

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We investigate scaling properties of а neighborhood of the «golden mean» critical point in the circle map in presence of external periodic forcing upon the system. We consider such perturbations of the fixed point of the Feigenbaum-Kadanoff-Shenker renormalization group equation, which are initiated by periodic forcing. We show that, depending upon the frequency of external forcing, two types of scaling behavior can be observed. The first (P-type) 18 associated with periodic repetition of the structures of dynamical regimes in parameter space under subsequent scaling transformations. For the second case (Q-type of scaling) quasiperiodic behavior of structures in the parameter space takes place. The both types of scaling are illustrated by parameter-space diagrams for re-scaled Lyapunov exponents.

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Yousefi, Saleh, Somayeh Mirzaee, Hussein Almohamad, et al. "Image Classification and Land Cover Mapping Using Sentinel-2 Imagery: Optimization of SVM Parameters." Land 11, no. 7 (2022): 993. http://dx.doi.org/10.3390/land11070993.

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Land use/cover (LU/LC) classification provides proxies of the natural and social processes related to urban development, providing stakeholders with crucial information. Remotely sensed images combined with supervised classification are common to define land use, but high-performance classifiers remain difficult to achieve, due to the presence of model hyperparameters. Conventional approaches rely on manual adjustment, which is time consuming and often unsatisfying. Therefore, the goal of this study has been to optimize the parameters of the support vector machine (SVM) algorithm for the generation of land use/cover maps from Sentinel-2 satellite imagery in selected humid and arid (three study sites each) climatic regions of Iran. For supervised SVM classification, we optimized two important parameters (gamma in kernel function and penalty parameter) of the LU/LC classification. Using the radial basis function (RBF) of the SVM classification method, we examined seven values for both parameters ranging from 0.001 to 1000. For both climate types, the penalty parameters (PP) showed a direct relationship with overall accuracy (OA). Statistical results confirmed that in humid study regions, LU/LC maps produced with a penalty parameter >100 were more accurate. However, for regions with arid climates, LU/LC maps with a penalty parameter >0.1 were more accurate. Mapping accuracy for both climate types was sensitive to the penalty parameter. In contrast, variations of the gamma values in the kernel function had no effect on the accuracy of the LU/LC maps in either of the climate zones. These new findings on SVM image classification are directly applicable to LU/LC for planning and environmental and natural resource management.

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