Weak Constraint Qualification Research Articles

Necessary and sufficient optimality conditions for nonsmooth nonconvex generalized Nash equilibrium problems

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.

Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization

In Haeser and Ramos (J Optim Theory Appl, 187:469–487, 2020), a generalization of the normal cone from single objective to multiobjective optimization is introduced, along with a weakest constraint qualification such that any local weak Pareto optimal point is a weak Kuhn–Tucker point. We extend this approach to other generalizations of the normal cone and corresponding weakest constraint qualifications, such that local Pareto optimal points are weak Kuhn–Tucker points, local proper Pareto optimal points are weak and proper Kuhn–Tucker points, respectively, and strict local Pareto optimal points of order one are weak, proper and strong Kuhn–Tucker points, respectively. The constructions are based on an appropriate generalization of polarity to pairs of matrices and vectors.

Hadamard well-posedness for a set optimization problem with an infinite number of constraints

This article aims to study the Hadamard well-posedness for a set optimization problem with an infinite number of constraints, where Kuroiwa's lower set less relation is used to compare entire images of an objective set-valued map. Based on the weak Slater constraint qualification, sufficient conditions of Hadamard well-posedness properties for efficient and weakly efficient solutions to the reference problem under functional perturbations of both constraint sets and objective maps are established. The obtained results can also be seen as an extension to the case of infinite/semi-infinite vector optimization problems.

Optimality and duality in nonsmooth semi-infinite optimization, using a weak constraint qualification

Optimality and duality in nonsmooth semi-infinite optimization, using a weak constraint qualification

Mordukhovich stationarity for mathematical programs with switching constraints under weak constraint qualifications

The mathematical program with switching constraints (MPSC), which has been introduced recently, is a difficult class of optimization problems since standard constraint qualifications are very likely to fail at local minimizers. Due to the failure of standard constraint qualifications, it is reasonable to propose some constraint qualifications for local minimizers to satisfy some stationarity conditions that are generally weaker than Karush-Kuhn-Tucker stationarity such as Mordukhovich (M-) stationarity. First, we propose the weakest constraint qualification for M-stationarity of MPSC to hold at local minimizers. Then we extend some weak verifiable constraint qualifications for nonlinear programming to allow the existence of switching constraints, which are all strictly weaker than MPSC linear independence constraint qualification and/or MPSC Mangasarian-Fromovitz constraint qualification used in the literature. We show that most of the newly introduced constraint qualifications are sufficient for local minimizers to be M-stationary. Finally, the relations among MPSC tailored constraint qualifications are discussed.

Generic Property of the Partial Calmness Condition for Bilevel Programming Problems

The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stationary conditions from a generic point of view. Our main result states that the partial error bound condition for the combined programs based on B and FJ conditions are generic for an important setting with applications in economics and hence the partial calmness for the combined program is not a particularly stringent assumption. Moreover we derive optimality conditions for the combined program for the generic case without any extra constraint qualifications and show the exact equivalence between our optimality condition and the one by Jongen and Shikhman given in implicit form. Our arguments are based on Jongen, Jonker and Twilt's generic (five type) classification of the so-called generalized critical points for one-dimensional parametric optimization problems and Jongen and Shikhman's generic local reductions of BLPPs.

Strong complementary approximate Karush-Kuhn-Tucker conditions for multiobjective optimization problems

In this paper, we establish strong complementary approximate Karush- Kuhn-Tucker (SCAKKT) sequential optimality conditions for multiobjective optimization problems with equality and inequality constraints without any constraint qualifications and introduce a weak constraint qualification which assures the equivalence between SCAKKT and the strong Karush-Kuhn-Tucker (J Optim Theory Appl 80 (3): 483-500, 1994) conditions for multiobjective optimization problems.

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

Parabolic regularity in geometric variational analysis

The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.

Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$-minimization problems

We consider an $$\ell _0$$ -minimization problem where $$f(x) + \gamma \Vert x\Vert _0$$ is minimized over a polyhedral set and the $$\ell _0$$ -norm regularizer implicitly emphasizes the sparsity of the solution. Such a setting captures a range of problems in image processing and statistical learning. Given the nonconvex and discontinuous nature of this norm, convex regularizers as substitutes are often employed and studied, but less is known about directly solving the $$\ell _0$$ -minimization problem. Inspired by Feng et al. (Pac J Optim 14:273–305, 2018), we consider resolving an equivalent formulation of the $$\ell _0$$ -minimization problem as a mathematical program with complementarity constraints (MPCC) and make the following contributions towards the characterization and computation of its KKT points: (i) First, we show that feasible points of this formulation satisfy the relatively weak Guignard constraint qualification. Furthermore, if f is convex, an equivalence is derived between first-order KKT points and local minimizers of the MPCC formulation. (ii) Next, we apply two alternating direction method of multiplier (ADMM) algorithms, named (ADMM $$_{\mathrm{cf}}^{\mu , \alpha , \rho }$$ ) and (ADMM $$_{\mathrm{cf}}$$ ), to exploit the special structure of the MPCC formulation. Both schemes feature tractable subproblems. Specifically, in spite of the overall nonconvexity, it is shown that the first update can be effectively reduced to a closed-form expression by recognizing a hidden convexity property while the second necessitates solving a tractable convex program. In (ADMM $$_{\mathrm{cf}}^{\mu , \alpha , \rho }$$ ), subsequential convergence to a perturbed KKT point under mild assumptions is proved. Preliminary numerical experiments suggest that the proposed tractable ADMM schemes are more scalable than their standard counterpart while (ADMM $$_{\mathrm{cf}}$$ ) compares well with its competitors in solving the $$\ell _0$$ -minimization problem.

Weak Slater Qualification for Nonconvex Multiobjective Semi-infinite Programming

We consider a nonsmooth multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints, MSIP for short. First, we introduce the weak Slater constraint qualification and derive the Karush–Kuhn–Tucker types necessary and sufficient conditions for (weakly, properly) efficient solution of the considered problem. Then, we introduce a dual of Mond–Weir type and present (weak and strong) duality results for MSIP. All results are given in terms of Clarke subdifferential.

An Augmented Lagrangian method for quasi-equilibrium problems

In this paper, we propose an Augmented Lagrangian algorithm for solving a general class of possible non-convex problems called quasi-equilibrium problems (QEPs). We define an Augmented Lagrangian bifunction associated with QEPs, introduce a secondary QEP as a measure of infeasibility and we discuss several special classes of QEPs within our theoretical framework. For obtaining global convergence under a new weak constraint qualification, we extend the notion of an Approximate Karush–Kuhn–Tucker (AKKT) point for QEPs (AKKT-QEP), showing that in general it is not necessarily satisfied at a solution, differently from its counterpart in optimization. We study some particular cases where AKKT-QEP does hold at a solution, while discussing the solvability of the subproblems of the algorithm. We also present illustrative numerical experiments.

Sufficiency and Duality for Interval-Valued Optimization Problems with Vanishing Constraints Using Weak Constraint Qualifications

In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem.

Mathematical programs with equilibrium constraints: a sequential optimality condition, new constraint qualifications and algorithmic consequences

ABSTRACT Mathematical programs with equilibrium constraints is a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications. Thus, the Karush–Kuhn–Tucker conditions are not necessarily satisfied at minimizers, and the convergence assumptions of many methods for solving constrained optimization problems are not fulfilled. Thus, it is necessary, from a theoretical and numerical point of view, to consider suitable optimality conditions, tailored constraints qualifications, and designed algorithms for solving such optimization problems. In this paper, we present a new sequential optimality condition useful for the convergence analysis of several methods for solving mathematical programs with equilibrium constraints such as relaxations schemes, complementarity-penalty methods, and interior-relaxation methods. Furthermore, the weakest constraint qualification for M-stationarity associated with such sequential optimality condition is presented. Relations between the old and new constraint qualifications, as well as the algorithmic consequences, will be discussed.

Two New Weak Constraint Qualifications for Mathematical Programs with Equilibrium Constraints and Applications

In this paper, we introduce two new constraint qualifications for mathematical programs with equilibrium constraints. One of them is a relaxed version of the No Nonzero Abnormal Multiplier Constraint Qualification, and the other is an adaptation of the Constant Rank of Subspace Component. The new conditions have nice properties. Indeed, they have the local preservation property and imply the error bound property under mild assumptions. Thus, they can be used to extend some known results on stability and sensitivity analysis. Furthermore, they can also be used in the convergence analysis of several methods for solving mathematical programs with equilibrium constraints.

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACTIn this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.

Optimality Conditions for Weakly ϵ-Efficient Solutions of Vector Optimization Problems with Applications

The aim of this article is to provide necessary and sufficient conditions for a feasible point to be a weakly ϵ-efficient solution of a vector optimization problem with cone constraints under generalized convexity notions and a weak constraint qualification. Our results permit to recover earlier established theorems.

On an exact penality result and new constraint qualifications for mathematical programs with vanishing constraints

In this paper, we considered the mathematical programs with vanishing constraints or MPVC. We proved that an MPVC-tailored penalty function, introduced in [5], is still exact under a very weak and new constraint qualification. Most importantly, this constraint qualification is shown to be strictly stronger than MPVC-Abadie constraint qualification.