Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints

In this paper, we consider a class of mathematical programs with switching constraints (MPSCs) where the objective involves a non-Lipschitz term. Due to the non-Lipschitz continuity of the objective function, the existing constraint qualifications for local Lipschitz MPSCs are invalid to ensure that necessary conditions hold at the local minimizer. Therefore, we propose some MPSC-tailored qualifications which are related to the constraints and the non-Lipschitz term to ensure that local minimizers satisfy the necessary optimality conditions. Moreover, we study the weak, Mordukhovich, Bouligand, strongly (W-, M-, B-, S-) stationay, analyze what qualifications making local minimizers satisfy the (M-, B-, S-) stationay, and discuss the relationship between the given MPSC-tailored qualifications. Finally, an approximation method for solving the non-Lipschitz MPSCs is given, and we show that the accumulation point of the sequence generated by the approximation method satisfies S-stationary under the second-order necessary condition and MPSC Mangasarian-Fromovitz (MF) qualification.

Inertial Extragradient Methods for Solving Split Equilibrium Problems

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

On implicit variables in optimization theory

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory. Comment: 34 pages

CONSTRAINT QUALIFICATIONS IN PARTIAL IDENTIFICATION

The literature on stochastic programming typically restricts attention to problems that fulfill constraint qualifications. The literature on estimation and inference under partial identification frequently restricts the geometry of identified sets with diverse high-level assumptions. These superficially appear to be different approaches to closely related problems. We extensively analyze their relation. Among other things, we show that for partial identification through pure moment inequalities, numerous assumptions from the literature essentially coincide with the Mangasarian–Fromowitz constraint qualification. This clarifies the relation between well-known contributions, including within econometrics, and elucidates stringency, as well as ease of verification, of some high-level assumptions in seminal papers.

On the convergence of augmented Lagrangian strategies for nonlinear programming

Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost.

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.

Relations between Abs-Normal NLPs and MPCCs. Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then we consider strong stationarity concepts with first and second order optimality conditions, which again turn out to be equivalent for the two problem classes. Throughout we also consider specific slack reformulations suggested in [9], which preserve constraint qualifications of linear independence type but not of Mangasarian-Fromovitz type.

Necessary optimality conditions in nonsmooth semi-infinite multiobjective optimization under metric subregularity

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 is extensive literature on standard semi-infinite programming problems. The investigation of optimality conditions for these problems is always one of the most attractive topics and has been studied extensively in the literature. In our work, we study optimality conditions for weak efficiency of a multiobjective semi-infinite optimization problem under mixed constraints including infinitely many of both equality and inequality constraints in terms of Clarke subdifferential. Our conditions are the form of the Karush-Kuhn-Tucker (KKT) multiplier. To the best of our knowledge, only a few papers are dealing with optimality conditions for SIPs subject to mixed constraints. By the Pshenichnyi-Levin-Valadire (PLV) property and the directional metric subregularity, we introduce a type of Mangasarian-Fromovitz constraint qualification (MFCQ). Then we show that (MFCQ) is a sufficient condition to guarantee the extended Abadie constraint qualification (ACQ) to satisfy. In our constraint qualifications, all functions are nonsmooth and the number of constraints is not necessarily finite. In our paper, we do not need the involved functions: convexity and differentiability. Later, we apply the extended Abadie constraint qualification to get the KKT multipliers for weak efficient solutions of SIP. Many examples are provided to illustrate some advantages of our results. The paper is organized as follows. In Section Preliminaries, we present our basic definitions of nonsmooth and convex analysis. Section Main Results prove necessary conditions for the weakly efficient solution in terms of the Karush-Kuhn-Tucker mult iplier rule with the help of some constraint qualifications.