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Journal articles on the topic 'Algorithms. Convex functions. Programming (Mathematics)'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Kebaili, Zahira, and Mohamed Achache. "Solving nonmonotone affine variational inequalities problem by DC programming and DCA." Asian-European Journal of Mathematics 13, no. 03 (December 17, 2018): 2050067. http://dx.doi.org/10.1142/s1793557120500679.

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In this paper, we consider an optimization model for solving the nonmonotone affine variational inequalities problem (AVI). It is formulated as a DC (Difference of Convex functions) program for which DCA (DC Algorithms) are applied. The resulting DCA are simple: it consists of solving successive convex quadratic program. Numerical experiments on several test problems illustrate the efficiency of the proposed approach in terms of the quality of the obtained solutions and the speed of convergence.

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2

Eckstein, Jonathan. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming." Mathematics of Operations Research 18, no. 1 (February 1993): 202–26. http://dx.doi.org/10.1287/moor.18.1.202.

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Awais, Hafiz Muhammad, Tahir Nadeem Malik, and Aftab Ahmad. "Artificial Algae Algorithm with Multi-Light Source Movement for Economic Dispatch of Thermal Generation." Mehran University Research Journal of Engineering and Technology 39, no. 3 (July 1, 2020): 564–82. http://dx.doi.org/10.22581/muet1982.2003.12.

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Economic Dispatch (ED) is one of the major concerns for the efficient and economical operation of the modern power system. Actual ED problem is non-convex in nature due to Ramp Rate Limits (RRL), Valve-Point Loading Effects (VPLE), and Prohibited Operating Zones (POZs). It is generally converted into a convex problem as mathematical programming based approaches cannot handle the non-convex cost functions except dynamic programming, which also suffers from the curse of dimensionality. Heuristic techniques are potential solution methodologies for solving the non-convex ED problem. Artificial Algae Algorithm (AAA), a recent meta-heuristic optimization approach showed remarkable results on certain MATLAB benchmark functions but its application on industrial problem such as ED is yet to be explored. In this paper, AAA is used to investigate convex and non-convex ED problem due to valve-point effects and POZs while considering the transmission losses. The robustness and effectiveness of the proposed approach are validated by implementing it on IEEE standard test systems (3, 6, 13 and 40 unit Test Systems), which are widely addressed in the literature. The simulation results are promising when compared with other well-known evolutionary algorithms, showing the potential and stability of this algorithm.

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Cocan, Moise, and Bogdana Pop. "An algorithm for solving the problem of convex programming with several objective functions." Korean Journal of Computational & Applied Mathematics 6, no. 1 (January 1999): 79–88. http://dx.doi.org/10.1007/bf02941908.

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5

Östermark, Ralf. "A parallel algorithm for optimizing the capital structure contingent on maximum value at risk." Kybernetes 44, no. 3 (March 2, 2015): 384–405. http://dx.doi.org/10.1108/k-08-2014-0171.

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Purpose – The purpose of this paper is to measure the financial risk and optimal capital structure of a corporation. Design/methodology/approach – Irregular disjunctive programming problems arising in firm models and risk management can be solved by the techniques presented in the paper. Findings – Parallel processing and mathematical modeling provide a fruitful basis for solving ultra-scale non-convex general disjunctive programming (GDP) problems, where the computational challenge in direct mixed-integer non-linear programming (MINLP) formulations or single processor algorithms would be insurmountable. Research limitations/implications – The test is limited to a single firm in an experimental setting. Repeating the test on large sample of firms in future research will indicate the general validity of Monte-Carlo-based VAR estimation. Practical implications – The authors show that the risk surface of the firm can be approximated by integrated use of accounting logic, corporate finance, mathematical programming, stochastic simulation and parallel processing. Originality/value – Parallel processing has potential to simplify large-scale MINLP and GDP problems with non-convex, multi-modal and discontinuous parameter generating functions and to solve them faster and more reliably than conventional approaches on single processors.

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Chao, Miantao, Yongxin Zhao, and Dongying Liang. "A Proximal Alternating Direction Method of Multipliers with a Substitution Procedure." Mathematical Problems in Engineering 2020 (April 27, 2020): 1–12. http://dx.doi.org/10.1155/2020/7876949.

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In this paper, we considers the separable convex programming problem with linear constraints. Its objective function is the sum of m individual blocks with nonoverlapping variables and each block consists of two functions: one is smooth convex and the other one is convex. For the general case m≥3, we present a gradient-based alternating direction method of multipliers with a substitution. For the proposed algorithm, we prove its convergence via the analytic framework of contractive-type methods and derive a worst-case O1/t convergence rate in nonergodic sense. Finally, some preliminary numerical results are reported to support the efficiency of the proposed algorithm.

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Dias, Bruno H., André L. M. Marcato, Reinaldo C. Souza, Murilo P. Soares, Ivo C. Silva Junior, Edimar J. de Oliveira, Rafael B. S. Brandi, and Tales P. Ramos. "Stochastic Dynamic Programming Applied to Hydrothermal Power Systems Operation Planning Based on the Convex Hull Algorithm." Mathematical Problems in Engineering 2010 (2010): 1–20. http://dx.doi.org/10.1155/2010/390940.

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This paper presents a new approach for the expected cost-to-go functions modeling used in the stochastic dynamic programming (SDP) algorithm. The SDP technique is applied to the long-term operation planning of electrical power systems. Using state space discretization, the Convex Hull algorithm is used for constructing a series of hyperplanes that composes a convex set. These planes represent a piecewise linear approximation for the expected cost-to-go functions. The mean operational costs for using the proposed methodology were compared with those from the deterministic dual dynamic problem in a case study, considering a single inflow scenario. This sensitivity analysis shows the convergence of both methods and is used to determine the minimum discretization level. Additionally, the applicability of the proposed methodology for two hydroplants in a cascade is demonstrated. With proper adaptations, this work can be extended to a complete hydrothermal system.

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SUN, YIJUN, SINISA TODOROVIC, and JIAN LI. "REDUCING THE OVERFITTING OF ADABOOST BY CONTROLLING ITS DATA DISTRIBUTION SKEWNESS." International Journal of Pattern Recognition and Artificial Intelligence 20, no. 07 (November 2006): 1093–116. http://dx.doi.org/10.1142/s0218001406005137.

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AdaBoost rarely suffers from overfitting problems in low noise data cases. However, recent studies with highly noisy patterns have clearly shown that overfitting can occur. A natural strategy to alleviate the problem is to penalize the data distribution skewness in the learning process to prevent several hardest examples from spoiling decision boundaries. In this paper, we pursue such a penalty scheme in the mathematical programming setting, which allows us to define a suitable classifier soft margin. By using two smooth convex penalty functions, based on Kullback–Leibler divergence (KL) and l2 norm, we derive two new regularized AdaBoost algorithms, referred to as AdaBoostKL and AdaBoostNorm2, respectively. We prove that our algorithms perform stage-wise gradient descent on a cost function, defined in the domain of their associated soft margins. We demonstrate the effectiveness of the proposed algorithms through experiments over a wide variety of data sets. Compared with other regularized AdaBoost algorithms, our methods achieve at least the same or better performance.

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Xu, Lei. "One-Bit-Matching Theorem for ICA, Convex-Concave Programming on Polyhedral Set, and Distribution Approximation for Combinatorics." Neural Computation 19, no. 2 (February 2007): 546–69. http://dx.doi.org/10.1162/neco.2007.19.2.546.

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According to the proof by Liu, Chiu, and Xu (2004) on the so-called one-bit-matching conjecture (Xu, Cheung, and Amari, 1998a), all the sources can be separated as long as there is an one-to-one same-sign correspondence between the kurtosis signs of all source probability density functions (pdf's) and the kurtosis signs of all model pdf's, which is widely believed and implicitly supported by many empirical studies. However, this proof is made only in a weak sense that the conjecture is true when the global optimal solution of an independent component analysis criterion is reached. Thus, it cannot support the successes of many existing iterative algorithms that usually converge at one of the local optimal solutions. This article presents a new mathematical proof that is obtained in a strong sense that the conjecture is also true when any one of local optimal solutions is reached in helping to investigating convex-concave programming on a polyhedral set. Theorems are also provided not only on partial separation of sources when there is a partial matching between the kurtosis signs, but also on an interesting duality of maximization and minimization on source separation. Moreover, corollaries are obtained on an interesting duality, with supergaussian sources separated by maximization and subgaussian sources separated by minimization. Also, a corollary is obtained to confirm the symmetric orthogonalization implementation of the kurtosis extreme approach for separating multiple sources in parallel, which works empirically but lacks mathematical proof. Furthermore, a linkage has been set up to combinatorial optimization from a distribution approximation perspective and a Stiefel manifold perspective, with algorithms that guarantee convergence as well as satisfaction of constraints.

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Popkov, Alexander S. "Optimal program control in the class of quadratic splines for linear systems." Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes 16, no. 4 (2020): 462–70. http://dx.doi.org/10.21638/11701/spbu10.2020.411.

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This article describes an algorithm for solving the optimal control problem in the case when the considered process is described by a linear system of ordinary differential equations. The initial and final states of the system are fixed and straight two-sided constraints for the control functions are defined. The purpose of optimization is to minimize the quadratic functional of control variables. The control is selected in the class of quadratic splines. There is some evolution of the method when control is selected in the class of piecewise constant functions. Conveniently, due to the addition/removal of constraints in knots, the control function can be piecewise continuous, continuous, or continuously differentiable. The solution algorithm consists in reducing the control problem to a convex mixed-integer quadratically-constrained programming problem, which could be solved by using well-known optimization methods that utilize special software.

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Tang, Chunming, Yanni Li, Xiaoxia Dong, and Bo He. "A Generalized Alternating Linearization Bundle Method for Structured Convex Optimization with Inexact First-Order Oracles." Algorithms 13, no. 4 (April 14, 2020): 91. http://dx.doi.org/10.3390/a13040091.

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In this paper, we consider a class of structured optimization problems whose objective function is the summation of two convex functions: f and h, which are not necessarily differentiable. We focus particularly on the case where the function f is general and its exact first-order information (function value and subgradient) may be difficult to obtain, while the function h is relatively simple. We propose a generalized alternating linearization bundle method for solving this class of problems, which can handle inexact first-order information of on-demand accuracy. The inexact information can be very general, which covers various oracles, such as inexact, partially inexact and asymptotically exact oracles, and so forth. At each iteration, the algorithm solves two interrelated subproblems: one aims to find the proximal point of the polyhedron model of f plus the linearization of h; the other aims to find the proximal point of the linearization of f plus h. We establish global convergence of the algorithm under different types of inexactness. Finally, some preliminary numerical results on a set of two-stage stochastic linear programming problems show that our method is very encouraging.

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Albers, A., H. Weiler, D. Emmrich, and B. Lauber. "A New Approach for Optimization of Sheet Metal Components." Advanced Materials Research 6-8 (May 2005): 255–62. http://dx.doi.org/10.4028/www.scientific.net/amr.6-8.255.

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Beads are a widespread technology for reinforcing sheet metal structures, because they can be applied without any additional manufacturing effort and without significant weight increase. The two main applications of bead technology are to increase the stiffness for static loading conditions and to reduce the noise and vibrations for dynamic loadings. However, it is difficult to design the bead patterns of sheet metal structures due to the direction-controlled reinforcement effect of the beads. A wrong bead pattern layout can even weaken the properties of the structure. In the past, the designs were predominantly determined empirically or by the use of so called bead catalogues. Recently, different optimization approaches for bead patterns were developed, which are based upon classical mathematical programming optimization algorithms together with automatically generated shape basis vectors. However, these approaches usually provide only vague suggestions for the designs. One of the most severe difficulty with these approaches is to transfer the optimized results into manufacturable designs. Furthermore, another severe difficulty is that the optimization problem is non-convex, which frequently leads the mathematical programming algorithms into a local optima and thus to sub-optimal solutions. The investigations in this article show an optimization method, which within a few iterations leads to bead structures with excellent reinforcement effects using optimality criteria based approach. Generally, the results can be transferred without large effort into a final design. The new optimization method calculates the distribution of the bending stress tensor and the principal bending stresses based upon the results of a finite element analysis. The bead orientations are calculated by the trajectories of the principal bending stress with the largest magnitude. The beads are projected on to the mesh of the component using geometric form functions of the desired bead cross section. A local bead ratio of 50% (defined as average area of the beads in relation to total area of the sheet) is used by the algorithm to determine the maximum moment of inertia. The proposed algorithm is numerical implemented in the optimization system TOSCA and available for being applied with the following finite element solvers: ABAQUS, ANSYS, I-DEAS, NX Nastran, MSC.Nastran, MSC.Marc and PERMAS. The optimization algorithm is successfully applied to static and dynamic real world problems like car body parts, oil pans and exhaust mufflers. In the present work several academic and industrial examples are presented.

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Lv, Zheng, Zhiping Qiu, and Qi Li. "An Interval Reduced Basis Approach and its Integrated Framework for Acoustic Response Analysis of Coupled Structural-Acoustic System." Journal of Computational Acoustics 25, no. 03 (September 2017): 1750009. http://dx.doi.org/10.1142/s0218396x17500096.

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An interval reduced basis approach (IRBA) is presented for analyzing acoustic response of coupled structural-acoustic system with interval parameters. Simultaneously an integrated framework based on IRBA is established to deal with uncertain acoustic propagation using deterministic finite element (FE) software. The present IRBA aims to improve the accuracy of the conventional first-order approximation and also allow the efficient calculation of second-order approximation of acoustic response. In IRBA, acoustic response is approximated using a linear combination of interval basis vectors with undetermined coefficients. To get explicit expression of acoustic response in terms of interval parameters, the three terms of the second-order perturbation method are employed as basis vectors, and the variant of the Galerkin scheme is applied for derivation of the reduced-order system of equations. For the second-order approximation, the determination of acoustic response interval is reformulated into a series of quadratic programming problems, which are solved using the difference of convex functions (DC) algorithm effectively. The performance of IRBA and availability of the present framework are validated by numerical examples.

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Tormena, F. V., E. G. F. Mercuri, and M. B. Hecke. "A Bone Remodelling Model Based on Generalised Thermodynamic Potentials and Optimisation Applied to a Trabecula with Cyclic Loading." Applied Bionics and Biomechanics 10, no. 4 (2013): 175–88. http://dx.doi.org/10.1155/2013/762867.

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Background: Bone diseases caused by an imbalance of bone turnover represent a major public health concern worldwide. Studies involving bone remodelling mechanisms can assist in the treatment of osteoporosis, osteopenia and in cases of fractures. In recent decades several authors have developed bone remodelling models.Aim: The aim of this study is to propose a model based on the thermodynamic framework to describe the process of bone remodelling. A secondary aim is to model a trabecula subjected to cyclic loading and calibrate the model with experimental data.Methods: Thermodynamic potentials are used to generate the functions of state based on internal scalar variables. The evolution of the variables in time is determined by dissipation potentials, which are created through the use of convex analysis. Constitutive equations are solved with mathematical programming algorithms and the numerical implementation of this theory uses the Finite Elements Method for spatial discretization.Results: The proposed theory was applied to a one-dimensional example, and two situations (an undamaged material and an initially damaged material) were simulated. The one-dimensional example shows a microscopic view of a trabecula under the influence of a growing load cycle throughout 1200 days. This dynamic process may represent the rehabilitation of an athlete, starting with light exercises up to a very heavy physical activity.Conclusions: The model was able to represent one bone remodelling cycle in the trabecula. Although it is not yet possible to obtain an experimental curve of a traction testin vivo, thein silicomodel showed a process of damage that is similar to the static test of the literature. The results also suggest a modification in the equation adopted for the Helmholtz potential shown here. This study presents a consistent thermodynamic formalism for bone remodelling, which may allow further contributions as the incorporation of chemical reactions, mass transference and anisotropic damage.

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15

Weir, T. "Programming with semilocally convex functions." Journal of Mathematical Analysis and Applications 168, no. 1 (July 1992): 1–12. http://dx.doi.org/10.1016/0022-247x(92)90185-g.

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Mukherjee, R. N., and S. K. Mishra. "Multiobjective Programming with Semilocally Convex Functions." Journal of Mathematical Analysis and Applications 199, no. 2 (April 1996): 409–24. http://dx.doi.org/10.1006/jmaa.1996.0150.

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17

Goldbach, R. "Some Randomized Algorithms for Convex Quadratic Programming." Applied Mathematics and Optimization 39, no. 1 (January 2, 1999): 121–42. http://dx.doi.org/10.1007/s002459900101.

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18

Yang, X. M. "On E-Convex Sets, E-Convex Functions, and E-Convex Programming." Journal of Optimization Theory and Applications 109, no. 3 (June 2001): 699–704. http://dx.doi.org/10.1023/a:1017532225395.

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Dolatnezhadsomarin, Azam, Esmaile Khorram, and Latif Pourkarimi. "Efficient algorithms for solving nonlinear fractional programming problems." Filomat 33, no. 7 (2019): 2149–79. http://dx.doi.org/10.2298/fil1907149d.

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In this paper, an efficient algorithm based on the Pascoletti-Serafini scalarization (PS) approach is proposed to obtain almost uniform approximations of the entire Pareto front of bi-objective optimization problems. Five test problems with convex, non-convex, connected, and disconnected Pareto fronts are applied to evaluate the quality of approximations obtained by the proposed algorithm. Results are compared with results of some algorithms including the normal constraint (NC), weighted constraint (WC), Benson type, differential evolution (DE) with binomial crossover, non-dominated sorting genetic algorithm-II (NSGA-II), and S metric selection evolutionary multiobjective algorithm (SMS-EMOA). The results confirm the effectiveness of the presented bi-objective algorithm in terms of the quality of approximations of the Pareto front and CPU time. In addition, two algorithms are presented for approximately solving fractional programming (FP) problems. The first algorithm is based on an objective space cut and bound method for solving convex FP problems and the second algorithm is based on the proposed bi-objective algorithm for solving nonlinear FP problems. In addition, several examples are provided to demonstrate the performance of these suggested fractional algorithms.

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Kozma, Attila, Christian Conte, and Moritz Diehl. "Benchmarking large-scale distributed convex quadratic programming algorithms." Optimization Methods and Software 30, no. 1 (May 12, 2014): 191–214. http://dx.doi.org/10.1080/10556788.2014.911298.

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21

Li, Min, Zhikai Jiang, and Zhangjin Zhou. "Dual–primal proximal point algorithms for extended convex programming." International Journal of Computer Mathematics 92, no. 7 (August 13, 2014): 1473–95. http://dx.doi.org/10.1080/00207160.2014.945920.

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Liu, Sanming, and Enmin Feng. "Another approach to multiobjective programming problems withF-convex functions." Journal of Applied Mathematics and Computing 17, no. 1-2 (March 2005): 379–90. http://dx.doi.org/10.1007/bf02936063.

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23

Stefanov, Stefan M. "Method for solving a convex integer programming problem." International Journal of Mathematics and Mathematical Sciences 2003, no. 44 (2003): 2829–34. http://dx.doi.org/10.1155/s0161171203210516.

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We consider a convex integer program which is a nonlinear version of the assignment problem. This problem is reformulated as an equivalent problem. An algorithm for solving the original problem is suggested which is based on solving the simple assignment problem via some of known algorithms.

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24

Alimohammady, Mohsen, Yeol Je Cho, Vahid Dadashi, and Mehdi Roohi. "Convex sub-differential sum rule via convex semi-closed functions with applications in convex programming." Applied Mathematics Letters 24, no. 8 (August 2011): 1289–94. http://dx.doi.org/10.1016/j.aml.2011.01.046.

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Trdlička, Jiří, and Zdeněk Hanzálek. "Distributed Algorithm for Real-Time Energy Optimal Routing Based on Dual Decomposition of Linear Programming." International Journal of Distributed Sensor Networks 8, no. 1 (September 26, 2011): 346163. http://dx.doi.org/10.1155/2012/346163.

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This work proposes a novel in-network distributed algorithm for real-time energy optimal routing in ad hoc and sensor networks for systems with linear cost functions and constant communication delays. The routing problem is described as a minimum-cost multicommodity network flow problem by linear programming and modified by network replication to a real-time aware form. Based on the convex programming theory we use dual decomposition to derive the distributed algorithm. Thanks to the exact mathematical derivation, the algorithm computes the energy optimal real-time routing. It uses only peer-to-peer communication between neighboring nodes and does not need any central node or knowledge about the whole network structure. Each node knows only the produced and collected data flow and the costs of its outgoing communication links. According to our knowledge, this work is the first, which solves the real-time routing problem with linear cost functions and constant communication delays, using the dual decomposition.

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Chan, Timothy M. "Deterministic Algorithms for 2-d Convex Programming and 3-d Online Linear Programming." Journal of Algorithms 27, no. 1 (April 1998): 147–66. http://dx.doi.org/10.1006/jagm.1997.0914.

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Le, Hoai Minh, Hoai An Le Thi, Tao Pham Dinh, and Van Ngai Huynh. "Block Clustering Based on Difference of Convex Functions (DC) Programming and DC Algorithms." Neural Computation 25, no. 10 (October 2013): 2776–807. http://dx.doi.org/10.1162/neco_a_00490.

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We investigate difference of convex functions (DC) programming and the DC algorithm (DCA) to solve the block clustering problem in the continuous framework, which traditionally requires solving a hard combinatorial optimization problem. DC reformulation techniques and exact penalty in DC programming are developed to build an appropriate equivalent DC program of the block clustering problem. They lead to an elegant and explicit DCA scheme for the resulting DC program. Computational experiments show the robustness and efficiency of the proposed algorithm and its superiority over standard algorithms such as two-mode K-means, two-mode fuzzy clustering, and block classification EM.

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Yin, Jing Ben, and Kun Li. "A Deterministic Method for a Class of Fractional Programming Problems with Coefficients." Key Engineering Materials 467-469 (February 2011): 531–36. http://dx.doi.org/10.4028/www.scientific.net/kem.467-469.531.

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The sum of linear fractional functions problem has attracted the interest of researchers and practitioners for a number of years. Since these types of optimization problems are non-convex, various specialized algorithms have been proposed for globally solving these problems. However, these algorithms are only for the case that sum of linear ratios problem without coefficients, and may be difficult to be solved. In this paper, a deterministic algorithm is proposed for globally solving the sum of linear fractional functions problem with coefficients. By utilizing an equivalent problem and linear relaxation technique, the initial non-convex programming problem is reduced to a sequence of linear relaxation programming problems. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems.

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Lai, H. C., and J. C. Liu. "On Minimax Fractional Programming of Generalized Convex Set Functions." Journal of Mathematical Analysis and Applications 244, no. 2 (April 2000): 442–65. http://dx.doi.org/10.1006/jmaa.2000.6715.

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Liao, Jiagen, and Tingsong Du. "On some characterizations of sub-b-s-convex functions." Filomat 30, no. 14 (2016): 3885–95. http://dx.doi.org/10.2298/fil1614885l.

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A new class of generalized convex functions called sub-b-s-convex functions is defined by modulating the definitions of s-convex functions and sub-b-convex functions. Similarly, a new class sub-bs-convex sets, which are generalizations of s-convex sets and sub-b-convex sets, is introduced. Furthermore, some basic properties of sub-b-s-convex functions in both general case and differentiable case are presented. In addition the sufficient conditions of optimality for both unconstrained and inequality constrained programming are established and proved under the sub-b-s-convexity.

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Weir, T., and V. Jeyakumar. "A class of nonconvex functions and mathematical programming." Bulletin of the Australian Mathematical Society 38, no. 2 (October 1988): 177–89. http://dx.doi.org/10.1017/s0004972700027441.

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A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.

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Lai, H. C., and J. C. Liu. "Complex Fractional Programming Involving Generalized Quasi/Pseudo Convex Functions." ZAMM 82, no. 3 (March 2002): 159–66. http://dx.doi.org/10.1002/1521-4001(200203)82:3<159::aid-zamm159>3.0.co;2-5.

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Skarin, V. D. "Barrier function method and correction algorithms for improper convex programming problems." Proceedings of the Steklov Institute of Mathematics 263, S2 (December 2008): 120–34. http://dx.doi.org/10.1134/s0081543808060126.

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Le Thi, Hoai An, Xuan Thanh Vo, and Tao Pham Dinh. "Efficient Nonnegative Matrix Factorization by DC Programming and DCA." Neural Computation 28, no. 6 (June 2016): 1163–216. http://dx.doi.org/10.1162/neco_a_00836.

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In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms—one computing all variables and one deploying a variable selection strategy—are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.

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YANG, XIAONAN, and HONG-KUN XU. "Projection algorithms for composite minimization." Carpathian Journal of Mathematics 33, no. 3 (2017): 389–97. http://dx.doi.org/10.37193/cjm.2017.03.14.

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Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms consist of two steps. Once the kth iterate is constructed, an inner circle of gradient descent process is executed through each local function, and then a parallel or cyclic projection process is applied to produce the (k + 1) iterate. These algorithms are proved to converge to an optimal solution of the composite minimization problem under investigation upon assuming boundedness of the gradients at the iterates of the local functions and the stepsizes being chosen appropriately.

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Fu, J. Y., and Y. H. Wang. "Arcwise Connected Cone-Convex Functions and Mathematical Programming." Journal of Optimization Theory and Applications 118, no. 2 (August 2003): 339–52. http://dx.doi.org/10.1023/a:1025451422581.

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Liu, G. S., J. Y. Han, and J. Z. Zhang. "Exact Penalty Functions for Convex Bilevel Programming Problems." Journal of Optimization Theory and Applications 110, no. 3 (September 2001): 621–43. http://dx.doi.org/10.1023/a:1017592429235.

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Khalafi, Delavar, and Bijan Davvaz. "Algebraic hyper-structures associated to convex analysis and applications." Filomat 26, no. 1 (2012): 55–65. http://dx.doi.org/10.2298/fil1201055k.

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In this paper, we generalize some concepts of convex analysis such as convex functions and linear functions on hyper-structures. Based on new definitions we obtain some important results in convex programming. A few suitable examples have been given for better understanding.

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Liu, Xiaoling, and Dehui Yuan. "Mathematical programming involving B-(Hp,r,α)-generalized convex functions." International Journal of Mathematical Analysis 15, no. 4 (2021): 167–79. http://dx.doi.org/10.12988/ijma.2021.912203.

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Monteiro, Renato D. C., and Ilan Adler. "Interior path following primal-dual algorithms. part II: Convex quadratic programming." Mathematical Programming 44, no. 1-3 (May 1989): 43–66. http://dx.doi.org/10.1007/bf01587076.

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Emama, Tarek. "Optimality for E-[0,1] convex multi-objective programming." Filomat 31, no. 3 (2017): 529–41. http://dx.doi.org/10.2298/fil1703529e.

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In this paper, we interest with deriving the sufficient and necessary conditions for optimal solution of special classes of Programming. These classes involve generalized E-[0,1] convex functions. Characterization of efficient solutions for E-[0,1] convex multi-objective Programming are obtained. Finally, sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are derived.

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42

Gardiner, Bryan, and Yves Lucet. "Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis." Set-Valued and Variational Analysis 18, no. 3-4 (August 25, 2010): 467–82. http://dx.doi.org/10.1007/s11228-010-0157-5.

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Ye, Yinyu. "Interior-Point Polynomial Algorithms in Convex Programming (Y. Nesterov and A. Nemirovskii)." SIAM Review 36, no. 4 (December 1994): 682–83. http://dx.doi.org/10.1137/1036175.

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Atallah, Mikhail J., and Michael T. Goodrich. "Parallel algorithms for some functions of two convex polygons." Algorithmica 3, no. 1-4 (November 1988): 535–48. http://dx.doi.org/10.1007/bf01762130.

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Skarin, V. D. "Approximation and regularization properties of augmented penalty functions in convex programming." Proceedings of the Steklov Institute of Mathematics 269, S1 (July 2010): 266–84. http://dx.doi.org/10.1134/s0081543810060222.

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Herzel1, Stefano, and Michael J. Todd. "Two interior-point algorithms for a class of convex programming problems." Optimization Methods and Software 5, no. 1 (January 1995): 27–55. http://dx.doi.org/10.1080/10556789508805601.

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Jia, Zehui, Ke Guo, and Xingju Cai. "Convergence Analysis of Alternating Direction Method of Multipliers for a Class of Separable Convex Programming." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/680768.

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The purpose of this paper is extending the convergence analysis of Han and Yuan (2012) for alternating direction method of multipliers (ADMM) from the strongly convex to a more general case. Under the assumption that the individual functions are composites of strongly convex functions and linear functions, we prove that the classical ADMM for separable convex programming with two blocks can be extended to the case with more than three blocks. The problems, although still very special, arise naturally from some important applications, for example, route-based traffic assignment problems.

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Shen, Feichao, Ying Zhang, and Xueyong Wang. "An Accelerated Proximal Algorithm for the Difference of Convex Programming." Mathematical Problems in Engineering 2021 (April 24, 2021): 1–9. http://dx.doi.org/10.1155/2021/9994015.

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In this paper, we propose an accelerated proximal point algorithm for the difference of convex (DC) optimization problem by combining the extrapolation technique with the proximal difference of convex algorithm. By making full use of the special structure of DC decomposition and the information of stepsize, we prove that the proposed algorithm converges at rate of O 1 / k 2 under milder conditions. The given numerical experiments show the superiority of the proposed algorithm to some existing algorithms.

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Gulati, T. R., and M. A. Islam. "Sufficiency and Duality in Multiobjective Programming Involving Generalized F-Convex Functions." Journal of Mathematical Analysis and Applications 183, no. 1 (April 1994): 181–95. http://dx.doi.org/10.1006/jmaa.1994.1139.

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Yokoyama, Kazunori. "ε-optimality criteria for convex programming problems via exact penalty functions." Mathematical Programming 56, no. 1-3 (August 1992): 233–43. http://dx.doi.org/10.1007/bf01580901.

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