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Strongly Stable Stationary Points for a Class of Generalized Equations
In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.
In this paper we consider the class of mathematical programs with complementarity constraints (MPCC). Under an appropriate constraint qualification of Mangasarian–Fromovitz type we present a topological and an equivalent algebraic characterization of a strongly stable C-stationary point for MPCC. Strong stability refers to the local uniqueness, existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. This concept of strong stability was originally introduced by Kojima for standard nonlinear optimization; here, its generalization to MPCC demands a sophisticated technique which takes the disjunctive properties of the solution set of MPCC into account.
Mathematical programs with equilibrium (or complementarity) constraints (MPECs) form a difficult class of optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications. Therefore, one typically applies specialized algorithms in order to solve MPECs. One very prominent class of specialized algorithms are the regularization (or relaxation) methods. The first regularization method for MPECs is due to Scholtes [SIAM J. Optim., 11 (2001), pp. 918-936], but in the meantime, there exist a number of different regularization schemes which try to relax the difficult constraints in different ways. However, almost all regularization methods converge to C-stationary points only, which is a very weak stationarity concept. An exception is a recent method by Kadrani, Dussault, and Benchakroun [SIAM J. Optim., 20 (2009), pp. 78-103], whose limit points are shown to be M-stationary. Here we provide a new regularization method which also converges to M-stationary points. The assumptions to prove this result are, in principle, significantly weaker than for all other relaxation schemes. Furthermore, our relaxed problem has a much more favorable geometric shape than the one proposed by Kadrani, Dussault, and Benchakroun.
Mathematical programs with equilibrium (or complementarity) constraints form a difficult class of optimization problems. The standard KKT conditions are not always necessary optimality conditions due to the fact that suitable constraint qualifications (CQs) are often violated. Alternatively, one can therefore use the Fritz John approach to derive necessary optimality conditions. While the usual Fritz John conditions do not provide much information, we prove an enhanced version of the Fritz John conditions. This version motivates the introduction of some new CQs which can then be used in order to obtain, for the first time, a completely elementary proof of the fact that a local minimum is an M-stationary point under one of these CQs. We also show how these CQs can be used to obtain a suitable exact penalty result under weaker or different assumptions than those that can be found in the literature.
. Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and su#cient optimality conditions involving the proximal coderivatives. As an illustration of applications, the result is applied to the bilevel programming problems where the lower level is a parametric linear quadratic problem. Key words. optimization problems, complementarity constraints, optimality conditions, bilevel programming problems, proximal normal cones AMS subject classifications. 49K99, 90C, 90D65 PII. S1052623497321882 1. Introduction. The main purpose of this paper is to derive necessary and su#cient optimality conditions for the optimization problem with complementarity constraints (OPCC) defined as follows: (OPCC) min f(x, y, u) s.t. #u, #(x, y, u)# = 0, u # 0, #(x, y, u) # 0 (1.1) L(x, y, u) = 0, g(x, y, u) ...
Mathematical programmes with disjunctive constraints (MPDCs for short) cover several different problem classes from nonlinear optimization including complementarity-, vanishing-, cardinality- and switching-constrained optimization problems. In this paper, we introduce an abstract but reasonable version of the prominent linear independence constraint qualification which applies to MPDCs. Afterwards, we derive first- and second-order optimality conditions for MPDCs under validity of this constraint qualification based on so-called strongly stationary points. Finally, we apply our findings to some popular classes of disjunctive programmes and compare the obtained results to those ones available in the literature. Particularly, new second-order optimality conditions for mathematical programmes with switching constraints are by-products of our approach.
We consider the class of mathematical problems with complementarity constraints (MPCC) and apply Kojima's concept of strongly stable stationary points (originally introduced for a standard optimization problem) to C-stationary points of MPCC under certain assumptions. This concept refers to local existence and uniqueness of a stationary point for each sufficiently small perturbed problem. Assuming that the number of active constraints is n+1 and an appropriate constraint qualification holds at the considered point, the goal of this paper is twofold: For MPCC we will present necessary conditions for strong stability as well as equivalent algebraic characterizations for this topological concept.
We consider the concept of strongly stable C-stationary points for mathematical programs with complementarity constraints. The original concept of strong stability was introduced by Kojima for standard optimization programs. Adapted to our context, it refers to the local existence and uniqueness of a C-stationary point for each sufficiently small perturbed problem. The goal of this paper is to discuss a Mangasarian-Fromovitz-type constraint qualification and, mainly, provide two conditions which are necessary for strong stability; one is another constraint qualification and the second one refers to bounds on the number of active constraints at the point under consideration.
We consider parameterized Mathematical Programs with Complementarity Constraints arising, e.g., in modeling of deregulated electricity markets. Using the standard rules of the generalized differential calculus we analyze qualitative stability of solutions to the respective M-stationarity conditions. In particular, we provide characterizations and criteria for the isolated calmness and the Aubin properties of the stationarity map. To this end, we introduce the second-order limiting coderivative of mappings and provide formulas for this notion and for the graphical derivative of the limiting coderivative in the case of the normal cone mapping to ℝn+.
This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
We show how first order optimality conditions for a very general nonlinear optimization problem may be derived in a conceptually simple and unified manner in terms of certain multivalued functions associated with the problem. Necessary conditions for general problems and sufficient conditions for convex problems are developed, and the classical multiplier conditions are shown to be related in a simple way to these.
The main subject of this book is perturbation analysis of continuous optimization problems. In the last two decades considerable progress has been made in that area, and it seems that it is time now to present a synthetic view of many important results that apply to various classes of problems.
In 1980, M. Kojima introduced the topological concept of strongly stable stationary solutions in smooth nonlinear optimization. In case that the Mangasarian–Fromovitz Constraint Qualification (MFCQ) is satisfied, he characterized strong stability with the aid of first- and second-order derivatives of the objective function and the constraint functions. In this article, we show that the MFCQ is, in fact, implied by the assumption of strong stability. †Dedicated to Diethard Pallaschke on the occasion of his 65th birthday.
When we construct mathematical models from practical problems in the field of operations research, economics, engineering, etc., the data which we can utilize usually have uncertainty. We may not get exact data or the data may be changing as time goes. In such a model it is important to take account of the stability of the solution. Here we say that a solution to a model is stable if any slight perturbation to the data yields a small change of the solution. In this paper we study stability of a mathematical programming model, which involves an objective function to be minimized (or maximized) under certain constraints. We give conditions on the data of the model which characterize the stability. Applications to a mathematical programming model having parameters and a class of computational methods are also discussed.
This paper considers generalized equations, which are convenient tools for formulating problems in complementarity and in mathematical programming, as well as variational inequalities. We introduce a regularity condition for such problems and, with its help, prove existence, uniqueness and Lipschitz continuity of solutions to generalized equations with parametric data. Applications to nonlinear programming and to other areas are discussed, and for important classes of such applications the regularity condition given here is shown to be in a certain sense the weakest possible condition under which the stated properties will hold
We consider optimization problems with a disjunctive structure of the feasible set. Using Guignard-type constraint qualifications
for these optimization problems and exploiting some results for the limiting normal cone by Mordukhovich, we derive different
optimality conditions. Furthermore, we specialize these results to mathematical programs with equilibrium constraints. In
particular, we show that a new constraint qualification, weaker than any other constraint qualification used in the literature,
is enough in order to show that a local minimum results in a so-called M-stationary point. Additional assumptions are also
discussed which guarantee that such an M-stationary point is in fact a strongly stationary point.
This survey is concerned with necessary and sufficient optimality conditions for smooth nonlinear programming problems with inequality and equality constraints. These conditions deal with strict local minimizers of order one and two and with isolated minimizers. In most results, no constraint qualification is required. The optimality conditions are formulated in such a way that the gaps between the necessary and sufficient conditions are small and even vanish completely under mild constraint qualifications.
The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely
many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima
for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed
problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification
we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives
of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied.
We study mathematical programs with complementarity constraints (MPCC). Special focus will be on C-stationary points. Under
the Linear Independence Constraint Qualification we characterize strong stability of C-stationary points (in the sense of
Kojima) by means of first and second order information of the defining functions. It turns out that strong stability of C-stationary
points allows a possible degeneracy of bi-active Lagrange multipliers. Some relations to other stationarity concepts (such
as A-, M-, S- and B-stationarity) are shortly discussed.
KeywordsMathematical programs with complementarity constraints–C-stationarity–Strong stability–Linear independence constraint qualification
Partitioned symmetric matrices, in particular the Hessian of the Lagrangian, play a fundamental role in nonlinear optimization. For this type of matrices S.-P. Han and O. Fujiwara recently presented an inertia theorem under a certain regularity assumption. We prove that this theorem is true without any regularity assumption. Then we consider matrix extensions preserving the sign of the determinant. Such extensions are shown to be related with the positive definiteness of some Schur complement. Under a regularity assumption this shows, from the viewpoint of linear algebra, the equivalence of strong stability in the sense of M. Kojima and strong regularity in the sense of S. M. Robinson. Finally, we discuss the inertia of a typical one-parameter family of symmetric matrices, occurring in various places in optimization (augmented Lagrangians, focal-point theory, etc.).
We study mathematical programs with complementarity constraints (MPCC) from a topological point of view. Special focus will be on C- stationary points. Under the Linear Independence Constraint Quali- cation (LICQ) we derive an equivariant Morse Lemma at nondegener- ate C-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Out- side the C-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. How- ever, when passing a C-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary C-index of the (nondegenerate) C-stationary point. The stationary C-index depends on both the restricted Hessian of the Lagrangian and the Lagrange multipliers related to bi-active complementarity constraints. Finally, some relations with other sta- tionarity concepts, such as W-, A-, M-, S- and B-stationarity, are discussed.
The feasible set of mathematical programs with complementarity constraints (MPCC) is considered. We discuss local stability of the feasible set with respect to perturbations (up to first order) of the defining functions. Here, stability refers to homeomorphy invariance under small perturbations. For stability we propose a kind of Mangasarian-Fromovitz condition (MFC) and its stronger version (SMFC). MFC is a natural constraint qualification for C-stationarity, and SMFC is a generalization of the well-known Clarke's maximal rank condition. It turns out that SMFC implies local stability. MFC and SMFC coincide in the case where the number of complementarity constraints (k) equals the dimension of the state space (n). Moreover, the equivalence of MFC and SMFC is also proven for the cases k=2 as well as under linear independence constraint qualification (LICQ) for MPCC.
This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.
The main goal of our study is an attempt to understand and classify nonsmooth structures arising within the optimization setting: P(f,F): min f(x) s.t. x in M[F], where f is a smooth real-valued objective function, F is a smooth vector-valued function and M[F] a feasible set defined by F in some structured way. We focus rather on the underlying nonsmooth structures which fit the smooth function F to define the feasible set M[F]. The basis of our study is the topological approach. It encompasses two objects: the feasible set M[F] and the lower level sets M[f, F]^a. These objects are considered according to topological, optimization and stability issues. On the topology and stability level we deal with topological invariants of M[F] and M[f, F]^a. Here the questionings mainly arise from. They lead to establishing of an adequate theory on the optimization level. For M[F] Lipschitz manifold property and so-called topological stability are discussed. They naturally lead to constraint qualifications for P(f,F). Topological changes of M[f, F]^a give rise to define stationary points and develop critical point theory for P(f,F). Each Chapter 2-5 is devoted to optimization problems with particular type of nonsmoothness: mathematical programming programs with complementarity constraints, general semi-infinite optimization problems, mathematical programming programs with vanishing constraints, bilevel optimization. For these problems above topological and stability issues are elaborated and corresponding optimization concepts are introduced. It is worth to point out that the same topological questionings provide different (analytical) optimization concepts while applied to particular problems. The difference between these analytically described optimization concepts is a key point in understanding and comparing different kinds of nonsmoothness. In Chapter 6 we enlighten the impacts of our topological approach on nonsmooth analysis theory. Topologically regular points of a min-type nonsmooth mappings F are introduced. The crucial property is that for topologically regular value y of F the nonempty set F^{-1}(y) is an n-l dimensional Lipschitz manifold. Corresponding nonsmooth versions of Sard's Theorem are given. Man betrachtet folgendes allgemeines Optimierungsproblem: P(f,F): min f(x) s.t. x in M[F], wobei f und F glatte Funktionen sind und M[F] die durch F definierte zulässige Menge bezeichnet. Die Nichtglattheit wird dadurch gegeben, dass F die Menge M[F] auf eine strukturierte Weise festlegt. Es werden nämlich vier Problemtypen untersucht: (i) Optimierungsprobleme mit Komplementaritätsnebenbedingungen (mathematical programming programs with complementarity constraints), (ii) Allgemeine Semi-Innite Optimierungsprobleme (general semi-infinite optimization problems), (iii) Optimierungsprobleme mit verschwindenden Nebenbedingungen (mathematical programming programs with vanishing constraints), (iv) Zweistuge Optimierungsprobleme (bilevel optimization). Das Hauptziel ist es, die nichtglatten Strukturen im Optimierungskontext topologisch zu untersuchen. Der topologische Zugang beinhaltet folgende Fragestellungen: (a) Unter welchen Bedingungen ist M[F] eine Lipschitz-Mannigfaltigkeit der passenden Dimension? (b) Unter welchen Bedingungen ist M[F] stabil, d.h. M[F] bleibt invariant bis auf Homoeomorphismus im Bezug auf glatte Störungen von F? (c) Wie ändert sich die Topologie der unteren Niveaumengen M[f, F]^a bis auf Homotopieaequivalenz? Es wird gezeigt, dass die Fragestellungen (a) und (b) zu Constraint Qualifications führen. Über (c) gelangt man zur Stationarität und zur Kritische-Punkte-Theorie im Sinne von Morse. Man bekommt neue topologisch relevante Optimierungskonzepte in Termen von Ableitungen der definierenden Funktionen f und F. Es ist wichtig anzumerken: die selben Fragestellungen (a)-(c) liefern verschiedene analytische Optimieriungskonzepte, wenn angewandt auf einzelne Problemtypen (i)-(iv). Genau der Unterschied zwischen diesen analytisch beschriebenen Optimierungskonzepten ist ein Schlüssel, die verschiedenen Typen der Nichtglattheit zu vergleichen und theoretisch zu verstehen. Darüber hinaus werden die Auswirkungen von (a) und (b) auf die Theorie der nichtglatten Analysis dargestellt. Es werden topologisch reguläre Punkte für nichtglatte Abbildungen F vom Minimum-Typ eingeführt. Die ausschlaggebende Eigenschaft ist, dass für topologisch reguläre Werte y von F die Menge F^{-1}(y) eine n-l dimensionale Lipschitz-Mannigfaltigkeit ist. Hier ist die Anwendung nichtglatter Versionen des Satzes über implizite Funktionen von Bedeutung (von Clarke bzw. Kummer). Es wird herausgearbeitet, dass die Schwierigkeit deren Anwendung darin besteht, eine passende Aufspaltung des R^n zu finden. Dies führt zum besseren Verständnis der nichtglatten Geometrie und Topologie. Entsprechende nichtglatte Versionen des Satzes von Sard werden bewiesen.