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The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP), which allows the strategies and the objective function of each player to depend on the decision variables of all other players. In this paper, we consider a GNEP where the objective and constraint functions are not necessarily differentiable or convex. By proposing an alternative approach based on the equivalence with a single constrained optimization problem, we then present new sharp KKT-type necessary optimality conditions for a solution of this problem under a weak constraint qualification. We also offer an example to show that these conditions are more efficient for such problems than the previous ones. Further under weak convexity assumptions, we prove that these necessary conditions are also sufficient for optimality. Finally, we extend the definitions of some constraint qualifications from classical optimization to GNEP and study their interrelations. Throughout the paper, several examples are provided to illustrate the results. © 2024 Informa UK Limited, trading as Taylor & Francis Group.
Optimization Letters (18624472)18(3)pp. 705-726
In this paper, a nonsmooth nonconvex robust optimization problem is considered. Using the idea of pseudo-differential, nonsmooth versions of the Robinson, Mangasarian–Fromovitz and Abadie constraint qualifications are introduced and their relations with the existence of a local error bound are investigated. Based on the pseudo-differential notion, new necessary optimality conditions are derived under the Abadie constraint qualification. Moreover, an example is provided to clarify the results. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.
Set-Valued and Variational Analysis (18770541)32(3)
In this paper, we investigate sequential M-stationarity conditions for a class of nonsmooth nonconvex general optimization problems. We introduce various types of such conditions and compare them with previously established conditions in smooth or convex cases. The application of the derived results is demonstrated in the context of nonsmooth sparsity-constrained optimization problems. Additionally, we devise a Lagrangian-type algorithm for a specific case of smooth sparsity problems. Several examples are presented throughout the paper to illustrate the results. © The Author(s), under exclusive licence to Springer Nature B.V. 2024.
Journal of Optimization Theory and Applications (00223239)199(2)pp. 856-861
In [1], some parts of the proofs of Theorems 3.1 and 3.2(ii) are needed to be revised. The conclusion of [1, Theorem 3.1] holds under an additional assumption, thus it is restated and proved. Also, the proof of [1, Theorem 3.2(ii)] is modified. With this method of proof, the USRC of the function (Formula presented.) at (Formula presented.) becomes a smaller set in comparison with its counterpart in [1, Theorem 3.2(ii)]. Thus we can say, the statement of [1, Theorem 3.2(ii)] is improved. Furthermore, [1, Lemmas 3.2 and 3.3] are not required and hence are omitted. Accordingly, the paragraph before these Lemmas is also removed. Further, [1, Theorem 3.3] gives a better form of the optimality condition and is updated. No other changes are required regarding the preliminaries, definitions, main conclusions and examples. First, we update [1, Theorem 3.1] by adding the following additional assumption from [2]: We say that the function F is calm at (Formula presented.) with some modulus (Formula presented.) , if there exists a positive scalar (Formula presented.) satisfying (Formula presented.) for each (Formula presented.) ([1, Theorem 3.1] updated) Assume that the function F is calm at (Formula presented.) with some modulus (Formula presented.) and (Formula presented.) is directionally differentiable at (Formula presented.). If EBCQ holds at (Formula presented.) with a constant (Formula presented.) , then ACQ is satisfied at (Formula presented.) with the same constant. (modified) Let (Formula presented.) (otherwise there is nothing to prove) and EBCQ be satisfied at (Formula presented.) with (Formula presented.). Assume also that (Formula presented.) (if (Formula presented.) , the ACQ obviously holds). Thus, there is a sequence (Formula presented.) such that (Formula presented.) The closedness of (Formula presented.) , gives us a sequence (Formula presented.) such that for each k, (Formula presented.) We assert that the sequence (Formula presented.) is bounded. Fixing (Formula presented.) and observing (1), we obtain the following inequalities for all k sufficiently large: (Formula presented.) which shows the boundedness of (Formula presented.) and the assertion is proved. Thus by passing to a subsequence, without relabelling, (Formula presented.) converges to some vector (Formula presented.). Now, By EBCQ one has (Formula presented.) Next, we claim that (Formula presented.) From [1, Lemma 3.1] and the fact that (Formula presented.) is directionally differentiable at (Formula presented.) we get (Formula presented.) which proves the claim. Now, it follows from (2), (3) and (1) that (Formula presented.) Using again [1, Lemma 3.1], the above especially implies that (Formula presented.) and completes the proof of the theorem. (Formula presented.) In what follows, the proof of [1, Theorem 3.2(ii)] is modified. By this modification, USRC of the function (Formula presented.) at (Formula presented.) becomes a smaller set and gives a better result; hence its statement is also improved. ([1, Theorem 3.2(ii)] updated) Assume that (Formula presented.) is an u.s.c. PJ of (Formula presented.) at (Formula presented.). Suppose also that (Formula presented.) and (Formula presented.) is a bounded USRC of (Formula presented.) at (Formula presented.). Then the closure of the set (Formula presented.) is an USRC of the function (Formula presented.) at (Formula presented.). (revised) Put (Formula presented.) and fix (Formula presented.). First, let us show that (Formula presented.) For given (Formula presented.) , we have (Formula presented.) Thus, using the definition of A, we get (Formula presented.) There are two possible cases: If (Formula presented.) , then trivially we obtain (Formula presented.) Hence, let (Formula presented.) If the following inequality holds: (Formula presented.) due to the cone property of (Formula presented.) , we get (Formula presented.) and the inequality in (4) holds trivially. Finally, the following case remains (Formula presented.) For each fixed (Formula presented.) one has (Formula presented.) Utilizing [1, Lemma 3.1], we have (Formula presented.) which means that (Formula presented.) , for all (Formula presented.) Now, since (Formula presented.) there exits some positive number c such that (Formula presented.). Thus for some sequence (Formula presented.) and for all k sufficiently large, one has (Formula presented.) Applying now the mean value Theorem in [1, Propostion 2.3], we have for each k, (Formula presented.) Using the upper semicontinuity of (Formula presented.) at (Formula presented.) , for given sequence (Formula presented.) , there exits (Formula presented.) satisfying (Formula presented.) Thus, there exists (Formula presented.) such that (Formula presented.) Choosing now subsequences (Formula presented.) and (Formula presented.) , and using the inequality in (6), we deduce that (Formula presented.) Observing that (Formula presented.) and taking limit as (Formula presented.) in the latter inequality, we arrive at the contradiction (Formula presented.) , which shows the case (Formula presented.) and the equality (5) do not occur together and the proof is completed. (Formula presented.) Since the USRC of the function (Formula presented.) is changed, the optimality condition in [1, Theorem 3.3] is improved and updated, accordingly. ([1, Theorem 3.3] updated) Suppose that ACQ is satisfied at the local optimal point (Formula presented.) of GOP. Let (Formula presented.) and (Formula presented.) are USRC and u.s.c. PJ of f and F at (Formula presented.) respectively and (Formula presented.) is a bounded USRC of (Formula presented.) at (Formula presented.). Then (Formula presented.) where (Formula presented.) is the positive constant of ACQ and l is the Lipschitz constant of the function f in a neighborhood of (Formula presented.). The proofs of [1, Theorems 3.1 and 3.2(ii)] are rectified and their statements are updated. Also, [1, Theorem 3.3] gives a better form of the optimality condition which is improved. © 2023, Springer Science+Business Media, LLC, part of Springer Nature.
Positivity (13851292)26(1)
In this paper, a nonsmooth mathematical program with second-order cone complementarity constraints (SOCMPCC) is studied. First scaled and strong scaled necessary optimality results are derived for a general nonsmooth optimization problem. Then the SOCMPCC is formulated as a general optimization problem whose constraint system is described by the graph of the projection operator over the second-order cone. Next exact expressions for the tangents and normals to this graph are established. Finally, the scaled M- and S-stationary necessary optimality results are verified for nonsmooth SOCMPCC under new weak scaled constraint qualifications. A number of examples are provided to illustrate the results obtained. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Optimization (10294945)71(10)pp. 2979-3005
In this paper, we consider the problem of minimizing a continuously differentiable function subject to sparsity constraints. We formulate this problem as an equivalent disjunctive constrained optimization program. Then, we extend some of the well-known constraint qualifications by using the contingent and normal cones of the sparsity set and show that these constraint qualifications can be applied to obtain the first-order optimality conditions. In addition, we give the first-order sufficient optimality conditions by defining a new generalized convexity notion. Furthermore, we present the second-order necessary and sufficient optimality conditions for sparsity constrained optimization problems. Finally, we provide some examples and special cases to illustrate the obtained results. © 2021 Informa UK Limited, trading as Taylor & Francis Group.
TOP (11345764)30(2)pp. 270-295
We focus on optimality conditions for an important class of nonconvex and nonsmooth optimization problems, where the objective and constraint functions are presented as a difference of two tangentially convex functions. The main contribution of this paper is to clarify several kinds of stationary solutions and their relations, and establish local optimality conditions with a nonconvex feasible set. Finally, several examples are given to illustrate the effectiveness of the obtained results. © 2021, Sociedad de Estadística e Investigación Operativa.
Positivity (13851292)25(2)pp. 701-729
In this paper, the robust approach (the worst case approach) for nonsmooth nonconvex optimization problems with uncertainty data is studied. First various robust constraint qualifications are introduced based on the concept of tangential subdifferential. Further, robust necessary and sufficient optimality conditions are derived in the absence of the convexity of the uncertain sets and the concavity of the related functions with respect to the uncertain parameters. Finally, the results are applied to obtain the necessary and sufficient optimality conditions for robust weakly efficient solutions in multiobjective programming problems. In addition, several examples are provided to illustrate the advantages of the obtained outcomes. © 2020, Springer Nature Switzerland AG.
Journal of Optimization Theory and Applications (00223239)186(1)pp. 86-101
A non-Lipschitz version of the Abadie constraint qualification is introduced for a nonsmooth and nonconvex general optimization problem. The relationship between the new Abadie-type constraint qualification and the local error bound property is clarified. Also, a necessary optimality condition, based on the pseudo-Jacobians, is derived under the Abadie constraint qualification. Moreover, some examples are given to illustrate the obtained results. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
Positivity (13851292)24(2)pp. 253-285
In this paper, the notion of graphical derivatives is applied to define a new class of several well-known constraint qualifications for a nonconvex multifunction M at a point of its graph. This class is called as “scaled constraint qualifications”. The reason of this terminology is that these conditions ensure the existence of bounded KKT multiplier vectors with a proper upper bound. The relations between these constraint qualifications and stability properties of M are also investigated. New sharp necessary optimality conditions with bounded multiplier vectors are derived for an optimization problem with a generalized equation constraint. The results are adapted to nonsmooth general constrained problems and nonsmooth mathematical programs with equilibrium constraints. © 2019, Springer Nature Switzerland AG.
Optimization Letters (18624472)13(5)pp. 1027-1038
This paper concerns a nonsmooth sparsity constrained optimization problem. We present first and second-order necessary and sufficient optimality conditions by using the concept of normal and tangent cones to the sparsity constraint set. Moreover, second-order tangent set to the sparsity constraint is described and then a new second-order necessary optimality condition is established. The results are illustrated by several examples. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
Numerical Functional Analysis and Optimization (15322467)40(16)pp. 1918-1938
We present optimality conditions for a class of nonsmooth and nonconvex constrained optimization problems. To achieve this aim, various well-known constraint qualifications are extended based on the concept of tangential subdifferential and the relations between them are investigated. Moreover, local and global necessary and sufficient optimality conditions are derived in the absence of convexity of the feasible set. In addition to the theoretical results, several examples are provided to illustrate the advantage of our outcomes. © 2019, © 2019 Taylor & Francis Group, LLC.
Numerical Functional Analysis and Optimization (15322467)39(1)pp. 11-37
In this paper, we study necessary optimality conditions for local Pareto and weak Pareto solutions of multiobjective problems involving inequality and equality constraints in terms of convexificators. We develop the enhanced Karush–Kuhn–Tucker conditions and introduce the associated pseudonormality and quasinormality conditions. We also introduce several other new constraint qualifications which entirely depend on the feasible set. Then a connecting link between these constraint qualifications is presented. Moreover, we provide several examples that clarify the interrelations between the different results that we have established. © 2018 Taylor & Francis.
Journal of Optimization Theory and Applications (00223239)179(3)pp. 778-799
A nonsmooth and nonconvex general optimization problem is considered. Using the idea of pseudo-Jacobians, nonsmooth versions of the Robinson and Mangasarian–Fromovitz constraint qualifications are defined and relationships between them and the local error bound property are investigated. A new necessary optimality condition, based on the pseudo-Jacobians, is derived under the local error bound constraint qualification. These results are applied for computing and estimating the Fréchet and limiting subdifferentials of value functions. Moreover, several examples are provided to clarify the obtained results. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Journal of Global Optimization (09255001)67(4)pp. 829-850
In this paper, a general optimization problem is considered to investigate the conditions which ensure the existence of Lagrangian vectors with a norm not greater than a fixed positive number. In addition, the nonemptiness and boundedness of the multiplier sets together with their exact upper bounds is characterized. Moreover, three new constraint qualifications are suggested that each of them follows a degree of boundedness for multiplier vectors. Several examples at the end of the paper indicate that the upper bound for Lagrangian vectors is easily computable using each of our constraint qualifications. One innovation is introducing the so-called bounded Lagrangian constraint qualification which is stated based on the nonemptiness and boundedness of all possible Lagrangian sets. An application of the results for a mathematical program with equilibrium constraints is presented. © 2016, Springer Science+Business Media New York.
Optimization (10294945)66(9)pp. 1445-1463
In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded. © 2017 Informa UK Limited, trading as Taylor & Francis Group.
Optimization Letters (18624472)10(1)pp. 63-76
The paper deals with with an optimization problem with a countably infinite system of Lipschitz equality constraints defined from an Asplund space to R. The main attention is paid to deriving new necessary optimality conditions in terms of stability conditions the constraint system. To do so, we reformulate the constraints to a generalized equation and we define a new weak constraint qualification based on the calmness property of multifunctions. © 2015, Springer-Verlag Berlin Heidelberg.
Set-Valued and Variational Analysis (18770541)24(3)pp. 483-497
In this paper, we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. We obtain necessary conditions of Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) types for a nonsmooth (MPEC) problem in terms of the lower Hadamard directional derivative. In particular sufficient conditions for MPECs are given where the involved functions have pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets is a class of generalized convex functions that include quasiconvex functions. © 2015, Springer Science+Business Media Dordrecht.
Optimization (10294945)65(1)pp. 67-85
In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular,we show that GS-stationary is the first-order optimality condition under generalized standardAbadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions. © 2014 Taylor & Francis.
Journal of Optimization Theory and Applications (00223239)160(2)pp. 415-438
In this paper, we pursue two goals. First, we find exact relationships between the three concepts of semismooth sets, functions, and maps. Then, we consider the nonsmooth calculus of these notions. Particularly, we prove that the Mordukhovich and linear subdifferentials (coderivatives) are equal for the semismooth functions (maps). Several examples are presented to illustrate the results of the paper. © 2013 Springer Science+Business Media New York.
Optimization Letters (18624472)7(2)pp. 361-373
We establish new characterizations of Asplund spaces in terms of conditions ensuring the calmness property for constraint set mappings and the validity of inverse image formula for a general constrained system. © 2011 Springer-Verlag.
Set-Valued and Variational Analysis (09276947)20(3)pp. 499-518
The paper deals with the calmness of two classes of nonconvex set-valued mappings in Asplund spaces and its application to equilibrium problems. Its main part is devoted to establish new sufficient conditions for calmness, which are derived in terms of coderivatives and w* boundaries of normal cones to constraint sets. In order to achieve this goal, a new concept so-called "sequential normal smoothness" for the sets in Asplund spaces is introduced and compared with two well-known notions of convexity and semismoothness. Finally, the results are applied to prove necessary optimality conditions for nonparametric equilibrium problems under new weak constraint qualifications. © 2012 Springer Science+Business Media B.V.
Applied Mathematics Letters (18735452)25(10)pp. 1354-1360
An extremal principle in the vein of Mordukhovich is proved based on the linear generalized gradient. © 2011 Elsevier Ltd. All rights reserved.
Nonlinear Analysis, Theory, Methods and Applications (0362546X)72(5)pp. 2694-2705
In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Then, we derive a necessary optimality result for nonsmooth MPEC on any Asplund space. Also, under generalized convexity assumptions, we establish sufficient optimality conditions for this program in Banach spaces. © 2009 Elsevier Ltd. All rights reserved.
Set-Valued and Variational Analysis (09276947)17(1)pp. 63-95
We study nonsmooth mathematical programs with equilibrium constraints. First we consider a general disjunctive program which embeds a large class of problems with equilibrium constraints. Then, we establish several constraint qualifications for these optimization problems. In particular, we generalize the Abadie and Guignard-type constraint qualifications. Subsequently, we specialize these results to mathematical program with equilibrium constraints. In our investigation, we show that a local minimum results in a so-called M-stationary point under a very weak constraint qualification. © 2009 Springer Science+Business Media B.V.
Journal of Convex Analysis (09446532)16(1)pp. 187-210
We consider a mathematical program with equilibrium constraints (MPBC). First we obtain a Lagrange multiplier rule based on the linear sub differential involving equality, inequality and set constraints. Then we propose new constraint qualifications for M-stationary condition to hold. Finally we establish the Fritz John and Karush-Kuhn Tucker M-stationary necessary conditions for a nonsmooth (MPBC) based on the Michel-Penot subdifferential. © Heldermann Verlag.
