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Journal articles on the topic 'Equality and inequality constraints'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

Revesz, Peter Z. "The Evaluation and the Computational Complexity of Datalog Queries of Boolean Constraint Databases." International Journal of Algebra and Computation 08, no. 05 (1998): 553–73. http://dx.doi.org/10.1142/s0218196798000260.

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In the database framework of Kanellakis et al. it was argued that constraint query languages should meet the closed-form requirement, that is, queries should take as input constraint databases and give as output constraint databases that use the same type of constraints. This paper shows that the closed-form requirement can be met for Datalog queries with Boolean equality constraints with double exponential time-complete data complexity, for Datalog queries with precedence and monotone inequality constraints in triple exponential-time data complexity. A closed-form evaluation is also shown for

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2

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 sequ

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3

Mullins, S. H., W. W. Charlesworth, and D. C. Anderson. "A New Method for Solving Mixed Sets of Equality and Inequality Constraints." Journal of Mechanical Design 117, no. 2A (1995): 322–28. http://dx.doi.org/10.1115/1.2826142.

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Presented is a general method for solving sets of nonlinear constraints that include inequalities. Inequality constraints are common in engineering design problems, such as kinematic synthesis. The proposed method uses a modified Newton’s method and introduces a slack variable and a slack constraint to convert each inequality into an equality constraint. Singular value decomposition is used to find the pseudo-inverse of the Jacobian at each iteration. Benefits of this method are that constraint scaling is not an issue and that the method often fails gracefully for inconsistent constraint sets

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4

Gupta, Rekha, and Manjari Srivastava. "Constraint qualifications in nonsmooth multiobjective optimization problem." Filomat 31, no. 3 (2017): 781–97. http://dx.doi.org/10.2298/fil1703781g.

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A multiobjective optimization problem (MOP) with inequality and equality constraints is considered where the objective and inequality constraint functions are locally Lipschitz and equality constraint functions are differentiable. Burachik and Rizvi [J. Optim. Theory Appl. 155, 477-491 (2012)] gave Guignard and generalized Abadie regularity conditions for a differentiable programming problem and derived Karush-Kuhn-Tucker (KKT) type necessary optimality conditions. In this paper, we have defined the nonsmooth versions of Guignard and generalized Abadie regularity conditions given by Burachik a

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5

Cheng, Haifang, Weilai Huang, and Jianhu Cai. "Solving a Fully Fuzzy Linear Programming Problem through Compromise Programming." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/726296.

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In the current literatures, there are several models of fully fuzzy linear programming (FFLP) problems where all the parameters and variables were fuzzy numbers but the constraints were crisp equality or inequality. In this paper, an FFLP problem with fuzzy equality constraints is discussed, and a method for solving this FFLP problem is also proposed. We first transform the fuzzy equality constraints into the crisp inequality ones using the measure of the similarity, which is interpreted as the feasibility degree of constrains, and then transform the fuzzy objective into two crisp objectives b

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6

Li, Mei Xia. "A Class of Augumented Lagrangian Function for Nonlinear Programming." Advanced Materials Research 271-273 (July 2011): 1955–60. http://dx.doi.org/10.4028/www.scientific.net/amr.271-273.1955.

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In this paper, we discuss an exact augumented Lagrangian functions for the non- linear programming problem with both equality and inequality constraints, which is the gen- eration of the augmented Lagrangian function in corresponding reference only for inequality constraints nonlinear programming problem. Under suitable hypotheses, we give the relation- ship between the local and global unconstrained minimizers of the augumented Lagrangian function and the local and global minimizers of the original constrained problem. From the theoretical point of view, the optimality solution of the nonline

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7

Husain, Iqbal, and Vikas K. Jain. "Nondifferentiable Multiobjective Programming with Equality and Inequality Constraints." Open Journal of Modelling and Simulation 01, no. 02 (2013): 7–13. http://dx.doi.org/10.4236/ojmsi.2013.12002.

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8

Das, A. K., R. Jana, and Deepmala. "Invex programming problems with equality and inequality constraints." Transactions of A. Razmadze Mathematical Institute 172, no. 3 (2018): 361–71. http://dx.doi.org/10.1016/j.trmi.2018.04.001.

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9

Feppon, F., G. Allaire, and C. Dapogny. "Null space gradient flows for constrained optimization with applications to shape optimization." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 90. http://dx.doi.org/10.1051/cocv/2020015.

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The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita (Math. Program. 18 (1980) 155–168) for equality constrained problems: the search direction is a combination of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is t

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10

Teo, K. L., and K. H. Wong. "Nonlinearly constrained optimal control problems." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 33, no. 4 (1992): 517–30. http://dx.doi.org/10.1017/s0334270000007207.

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AbstractIn a paper by Teo and Jennings, a constraint transcription is used together with the concept of control parametrisation to devise a computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type. The aim of this paper is to extend the results to a more general class of constrained optimal control problems, where the problem is also subject to terminal equality state constraints. For illustration, a numerical example is included.

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11

Patzi Aquino, Paola Geovanna, M. d. R. de Pinho, and Geraldo Nunes Silva. "Necessary optimality conditions for minimax optimal control problems with mixed constraints." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 72. http://dx.doi.org/10.1051/cocv/2021069.

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A weak maximal principle for minimax optimal control problems with mixed state-control equality and inequality constraints is provided. In the formulation of the minimax control problem we allow for parameter uncertainties in all functions involved: in the cost function, in the dynamical control system and in the equality and inequality constraints. Then a new constraint qualification of Mangassarian-Fromovitz type is introduced which allowed us to prove the necessary conditions of optimality. We also derived the optimality conditions under a full rank conditions type and showed that it is, as

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12

Ronagh, Pooya, Brad Woods, and Ehsan Iranmanesh. "Solving constrained quadratic binary problems via quantum adiabatic evolution." Quantum Information and Computation 16, no. 11&12 (2016): 1029–47. http://dx.doi.org/10.26421/qic16.11-12-6.

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Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bou

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13

Oh, Man-Suk. "Bayesian comparison of models with inequality and equality constraints." Statistics & Probability Letters 84 (January 2014): 176–82. http://dx.doi.org/10.1016/j.spl.2013.10.005.

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14

Minghui Zhu and S. Martinez. "On Distributed Convex Optimization Under Inequality and Equality Constraints." IEEE Transactions on Automatic Control 57, no. 1 (2012): 151–64. http://dx.doi.org/10.1109/tac.2011.2167817.

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15

Ginchev, Ivan, Angelo Guerraggio, and Matteo Rocca. "Locally Lipschitz vector optimization with inequality and equality constraints." Applications of Mathematics 55, no. 1 (2010): 77–88. http://dx.doi.org/10.1007/s10492-010-0003-y.

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16

Li, Mingwu, and Harry Dankowicz. "Optimization with equality and inequality constraints using parameter continuation." Applied Mathematics and Computation 375 (June 2020): 125058. http://dx.doi.org/10.1016/j.amc.2020.125058.

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17

Tsao, H. S. Jacob, and Shu-Cherng Fang. "Linear programming with inequality constraints via entropic perturbation." International Journal of Mathematics and Mathematical Sciences 19, no. 1 (1996): 177–84. http://dx.doi.org/10.1155/s0161171296000257.

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A dual convex programming approach to solving linear programs with inequality constraints through entropic perturbation is derived. The amount of perturbation required depends on the desired accuracy of the optimum. The dual program contains only non-positivity constraints. Anϵ-optimal solution to the linear program can be obtained effortlessly from the optimal solution of the dual program. Since cross-entropy minimization subject to linear inequality constraints is a special case of the perturbed linear program, the duality result becomes readily applicable. Many standard constrained optimiza

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18

Al-hawasy, Jamil Amir, and Safaa J. Mohammed Al-Qaisi. "The Continuous Classical Boundary Optimal Control of a Couple Nonlinear Elliptic Partial Differential Equations with State Constraints." Al-Mustansiriyah Journal of Science 30, no. 1 (2019): 143. http://dx.doi.org/10.23851/mjs.v30i1.464.

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This paper is concerned with, the proof of the existence and the uniqueness theorem for the solution of the state vector of a couple of nonlinear elliptic partial differential equations using the Minty-Browder theorem, where the continuous classical boundary control vector is given. Also the existence theorem of a continuous classical boundary optimal control vector governing by the couple of nonlinear elliptic partial differential equation with equality and inequality constraints is proved. The existence of the uniqueness solution of the couple of adjoint equations associated with the conside

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19

Wolak, Frank A. "Local and Global Testing of Linear and Nonlinear Inequality Constraints in Nonlinear Econometric Models." Econometric Theory 5, no. 1 (1989): 1–35. http://dx.doi.org/10.1017/s0266466600012238.

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This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints test H: h(β) ≥ 0 versus K: β ∈ RK. We extend these results to consider local hypotheses tests of the form H: h1(β) ≥ 0 and h2(β) = 0 versus K: β ∈ RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributio

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20

Abur, A., and M. K. Celik. "Least absolute value state estimation with equality and inequality constraints." IEEE Transactions on Power Systems 8, no. 2 (1993): 680–86. http://dx.doi.org/10.1109/59.260812.

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21

Gao, Yan, and Defeng Sun. "Calibrating Least Squares Semidefinite Programming with Equality and Inequality Constraints." SIAM Journal on Matrix Analysis and Applications 31, no. 3 (2010): 1432–57. http://dx.doi.org/10.1137/080727075.

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22

Xu, Qing, Xi Dai, and Bo Yu. "Solving generalized Nash equilibrium problem with equality and inequality constraints." Optimization Methods and Software 24, no. 3 (2009): 327–37. http://dx.doi.org/10.1080/10556780802578884.

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23

Strekalovsky, Alexander S. "On Nonconvex Optimization Problems with D.C. Equality and Inequality Constraints." IFAC-PapersOnLine 51, no. 32 (2018): 895–900. http://dx.doi.org/10.1016/j.ifacol.2018.11.431.

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24

Jabr, R. A. "Power system Huber M-estimation with equality and inequality constraints." Electric Power Systems Research 74, no. 2 (2005): 239–46. http://dx.doi.org/10.1016/j.epsr.2004.11.002.

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25

Andreani, Roberto, Valeriano Antunes de Oliveira, Jamielli Tomaz Pereira, and Geraldo Nunes Silva. "A weak maximum principle for optimal control problems with mixed constraints under a constant rank condition." IMA Journal of Mathematical Control and Information 37, no. 3 (2020): 1021–47. http://dx.doi.org/10.1093/imamci/dnz036.

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Abstract Necessary optimality conditions for optimal control problems with mixed state-control equality constraints are obtained. The necessary conditions are given in the form of a weak maximum principle and are obtained under (i) a new regularity condition for problems with mixed linear equality constraints and (ii) a constant rank type condition for the general non-linear case. Some instances of problems with equality and inequality constraints are also covered. Illustrative examples are presented.

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26

Li, Guoquan, and Yan Wang. "Global Optimality Conditions for Nonlinear Programming Problems with Linear Equality Constraints." Journal of Applied Mathematics 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/213178.

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Some necessary global optimality conditions and sufficient global optimality conditions for nonconvex minimization problems with a quadratic inequality constraint and a linear equality constraint are derived. In particular, global optimality conditions for nonconvex minimization over a quadratic inequality constraint which extend some known global optimality conditions in the existing literature are presented. Some numerical examples are also given to illustrate that a global minimizer satisfies the necessary global optimality conditions but a local minimizer which is not global may fail to sa

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27

Agrawal, R., G. L. Kinzel, R. Srinivasan, and K. Ishii. "Engineering Constraint Management Based on an Occurrence Matrix Approach." Journal of Mechanical Design 115, no. 1 (1993): 103–9. http://dx.doi.org/10.1115/1.2919305.

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In many mechanical systems, the mathematical model can be characterized by m nonlinear equations in n unknowns. The m equations could be either equality constraints or active inequality constraints in a constrained optimization framework. In either case, the mathematical model consists of (n-m) degrees of freedom, and (n-m) unknowns must be specified before the system can be analyzed. In the past, designers have often fixed the set of (n-m) specification variables and computed the remaining n variables using the n equations. This paper presents constraint management algorithms that give the de

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28

Bhattacharya, Joydeep, Xue Qiao, and Min Wang. "ENDOGENOUS BORROWING CONSTRAINTS AND WEALTH INEQUALITY." Macroeconomic Dynamics 20, no. 6 (2016): 1413–31. http://dx.doi.org/10.1017/s1365100514000959.

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This paper studies the evolution of wealth inequality in an economy with endogenous borrowing constraints. In the model economy, young agents need to borrow to finance human capital investments but cannot commit to repaying their loans. Creditors can punish defaulters by banishing them permanently from the credit market. At equilibrium, loan default is prevented by imposing a borrowing limit tied to the borrower's inheritance. The heterogeneity in inheritances translates into heterogeneity in borrowing limits: endogenously, some borrowers face a zero borrowing limit, and some are partly constr

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29

Giorgi, Giorgio. "Minimum Principle-Type Necessary Optimality Conditions in Scalar and Vector Optimization. An Account." Journal of Mathematics Research 9, no. 4 (2017): 168. http://dx.doi.org/10.5539/jmr.v9n4p168.

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We take into condideration necessary optimality conditions of minimum principle-type, that is for optimization problems having, besides the usual inequality and/or equality constraints, a set constraint. The first part pf the paper is concerned with scalar optimization problems; the second part of the paper deals with vector optimization problems.

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30

Kumar, Ajith, and William R. Dillon. "Constrained Discrimination via MDI Estimation: The use of Additional Information in Segmentation Analysis." Journal of Marketing Research 24, no. 4 (1987): 396–403. http://dx.doi.org/10.1177/002224378702400407.

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The authors demonstrate a general, flexible constrained discrimination method for testing hypotheses about the segmentability of a target population using categorical descriptors when additional information is available. The method applies the principle of minimum discrimination information (MDI) to the estimation of multinomial probabilities under linear equality and inequality constraints.

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31

Uppal, Tarun, Soumyendu Raha, and Suresh Srivastava. "Inverse Simulation for Gas Turbine Engine Control through Differential Algebraic Inequality Formulation." International Journal of Turbo & Jet-Engines 35, no. 4 (2018): 373–83. http://dx.doi.org/10.1515/tjj-2016-0057.

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Abstract Modern day gas turbines are prime movers in land, air and sea. They have stringent performance requirements to meet the complex mission objectives. Optimal control strategies can help them meet their performance objectives more efficiently. A novel inverse simulation method for optimal control and system analysis studies using Differential Algebraic Equality/Inequality (DAE/DAI) technique is brought out in this paper with a case study. The gas turbine model together with safety constraints and performance specifications is represented as a high index DAI/DAE system. The solution for t

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32

Chen, Yong, and Bailin Li. "A Modification of Feasible Direction Optimization Method for Handling Equality Constraints." Journal of Mechanisms, Transmissions, and Automation in Design 111, no. 3 (1989): 442–45. http://dx.doi.org/10.1115/1.3259018.

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The Feasible Direction Method of Zoutendijk has proven to be one of the most efficient algorithms currently available for solving nonlinear programming problems with only inequality type constraints. Since in the case of equality type constraints, there exists no nonzero direction close to the feasible region, the traditional algorithm cannot work properly. In this paper, a modified feasible direction finding technique has been proposed in order to handle nonlinear equality constraints for the Feasible Direction Method. The algorithm is based on searching along directions intersecting the tang

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33

Kawasaki, Hidefumi, and Vera Zeidan. "Conjugate Points for Variational Problems with Equality and Inequality State Constraints." SIAM Journal on Control and Optimization 39, no. 2 (2000): 433–56. http://dx.doi.org/10.1137/s0363012998345925.

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34

Husain, Iqbal, and Santosh K. Shrivastav. "Fritz John Duality in the Presence of Equality and Inequality Constraints." Applied Mathematics 03, no. 09 (2012): 1023–28. http://dx.doi.org/10.4236/am.2012.39151.

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35

Ledzewicz, U. "On Abnormal Optimal Control Problems with Mixed Equality and Inequality Constraints." Journal of Mathematical Analysis and Applications 173, no. 1 (1993): 18–42. http://dx.doi.org/10.1006/jmaa.1993.1050.

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36

Liu, Xin-wei, and Ya-xiang Yuan. "A Robust Algorithm for Optimization with General Equality and Inequality Constraints." SIAM Journal on Scientific Computing 22, no. 2 (2000): 517–34. http://dx.doi.org/10.1137/s1064827598334861.

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37

Soest, Arthur van, Arie Kapteyn, and Peter Kooreman. "Coherency and regularity of demand systems with equality and inequality constraints." Journal of Econometrics 57, no. 1-3 (1993): 161–88. http://dx.doi.org/10.1016/0304-4076(93)90063-b.

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38

Yin, Hui, Ye-Hwa Chen, and Dejie Yu. "Vehicle motion control under equality and inequality constraints: a diffeomorphism approach." Nonlinear Dynamics 95, no. 1 (2018): 175–94. http://dx.doi.org/10.1007/s11071-018-4558-6.

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39

Zhou, Zhong, Anthony Chen, and Deren Han. "An extended alternating direction method for variational inequality problems with linear equality and inequality constraints." Applied Mathematics and Computation 184, no. 2 (2007): 769–82. http://dx.doi.org/10.1016/j.amc.2006.05.205.

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40

Эглит, Я. Я., К. Я. Эглите, А. А. Ковтун, and Д. А. Глушко. "Mathematical relations for constructing an algorithm for estimating signal parameters under conditions of constraints." MORSKIE INTELLEKTUAL`NYE TEHNOLOGII), no. 2(52) (June 20, 2021): 153–56. http://dx.doi.org/10.37220/mit.2021.52.2.085.

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Статья посвящена разработке математических соотношений для построения алгоритма оценивания параметров сигналов в условиях ограничений. При работе транспортной системы возникают довольно сложные проблемы, которые связаны с необходимостью проведения оценки принятых параметров с требованиями соблюдения имеющихся ограничений. Ограничения могут представлять собой как равенства, так и неравенства. Поскольку ограничения-неравенства могут быть сведены путём добавления фиктивных переменных к условиям, а также их можно проверить по шагам, переводя в состав равенства, в статье разработан алгоритм, позвол

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41

Song, Hyoung Yong, Yong Gi Park, Jae Hyung Roh, Jong Bae Park, and Joong Rin Shin. "Immune-PSO for Economic Dispatch with Valve Point Effect." Advanced Materials Research 452-453 (January 2012): 1054–58. http://dx.doi.org/10.4028/www.scientific.net/amr.452-453.1054.

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This paper presents a simple and efficient approach for solving the nonconvex economic dispatch (ED) problems based on Immune-pso algorithm. ED is one of the most important problems in power system operation and planning. The aim of the ED problem is to determine the optimal combination of power outputs of all generating units so as to meet the required load demand at minimum operating cost while satisfying system equality and inequality constraints. This paper proposes an novel Immune-PSO algorithm to handle the equality and inequality constraints in ED problems. The Immune-PSO can be used as

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42

Gonggui, Chen, Huang Shanwai, and Sun Zhi. "A Chaotic Quantum Behaved Particle Swarm Optimization Algorithm for Short-term Hydrothermal Scheduling." Open Electrical & Electronic Engineering Journal 11, no. 1 (2017): 23–37. http://dx.doi.org/10.2174/1874129001711010023.

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This study proposes a novel chaotic quantum-behaved particle swarm optimization (CQPSO) algorithm for solving short-term hydrothermal scheduling problem with a set of equality and inequality constraints. In the proposed method, chaotic local search technique is employed to enhance the local search capability and convergence rate of the algorithm. In addition, a novel constraint handling strategy is presented to deal with the complicated equality constrains and then ensures the feasibility and effectiveness of solution. A system including four hydro plants coupled hydraulically and three therma

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43

Nguyen Dinh, Huy, Tinh Cao Thanh, Tung Nguyen, and Oanh Cao Thi Be. "Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity." Science & Technology Development Journal - Engineering and Technology 3, SI3 (2021): SI52—SI57. http://dx.doi.org/10.32508/stdjet.v3isi3.637.

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We consider nonsmooth semi-infinite multiobjective optimization problems under mixed constraints, including infinitely many mixed constraints by using Clarke subdifferential. Semi-infinite programming (SIP) is the minimization of many scalar objective functions subject to a possibly infinite system of inequality or/and equality constraints. SIPs have been proved to be very important in optimization and applications. Semi-infinite programming problems arise in various fields of engineering such as control systems design, decision making under competition, and multiobjective optimization. There

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44

Su, Menglong, Yufeng Shang, and Wenzhuang Zhu. "A Parameter Perturbation Homotopy Continuation Method for Solving Fixed Point Problems with Both Inequality and Equality Constraints." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/4067202.

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In this paper, we propose a parameter perturbation homotopy continuation method for solving fixed point problems on more general nonconvex sets with both inequality and equality constraints. By adopting appropriate techniques, we make the initial points not certainly in the set consisting of the equality constraints. This point can improve the computational efficiency greatly when the equality constraints are complex. In addition, we also weaken the assumptions of the previous results in the literature so that the method proposed in this paper can be applied to solve fixed point problems in mo

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45

Singh, O. V., and M. Singh. "A Comparative Analysis on Economic Load Dispatch Problem Using Soft Computing Techniques." International Journal of Software Science and Computational Intelligence 12, no. 2 (2020): 50–73. http://dx.doi.org/10.4018/ijssci.2020040104.

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This article aims at solving economic load dispatch (ELD) problem using two algorithms. Here in this article, an implementation of Flower Pollination (FP) and the BAT Algorithm (BA) based optimization search algorithm method is applied. More than one objective is hoped to be achieve in this article. The combined economic emission dispatch (CEED) problem which considers environmental impacts as well as the cost is also solved using the two algorithms. Practical problems in economic dispatch (ED) include both nonsmooth cost functions having equality and inequality constraints which make it diffi

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46

Xiaona Fan, and Furong Gao. "Solving Variational Inequalities with Equality and Inequality Constraints on an Unbounded Set." INTERNATIONAL JOURNAL ON Advances in Information Sciences and Service Sciences 5, no. 11 (2013): 264–72. http://dx.doi.org/10.4156/aiss.vol5.issue11.32.

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47

Treiman, Jay S. "Lagrange Multipliers for Nonconvex Generalized Gradients with Equality, Inequality, and Set Constraints." SIAM Journal on Control and Optimization 37, no. 5 (1999): 1313–29. http://dx.doi.org/10.1137/s0363012996306595.

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48

Persson, H., and Referee M. Clarke. "Algorithm 12: Solving the Entropy Maximization Problem with Equality and Inequality Constraints." Environment and Planning A: Economy and Space 18, no. 12 (1986): 1665–76. http://dx.doi.org/10.1068/a181665.

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49

Arutyunov, A. V., D. Y. Karamzin, and F. L. Pereira. "Necessary Optimality Conditions for Problems with Equality and Inequality Constraints: Abnormal Case." Journal of Optimization Theory and Applications 140, no. 3 (2008): 391–408. http://dx.doi.org/10.1007/s10957-008-9456-5.

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50

Ito, K., and K. Kunisch. "The augmented lagrangian method for equality and inequality constraints in hilbert spaces." Mathematical Programming 46, no. 1-3 (1990): 341–60. http://dx.doi.org/10.1007/bf01585750.

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