Partial Calmness Research Articles

Necessary optimality conditions for strictly robust bilevel optimization problems

The study of robust bilevel programming problems is a relatively new area of optimization theory. In this work, we investigate a bilevel optimization problem where the upper-level and the lower-level constraints incorporate uncertainty. Reducing the problem into a single-level nonlinear and nonsmooth program, necessary optimality conditions are then developed in terms of Clarke subdifferentials. Our approach consists of using the optimal value reformulation together with a partial calmness condition for the robust counterpart of the initial problem. To aid in the detection of Karush-Kuhn-Tucker (KKT) multipliers, an appropriate nonsmooth Mangasarian-Fromovitz constraint qualification is introduced. There are examples highlighting both our results and the limits of certain past studies.

On the Partial Calmness Condition for an Interval-Valued Bilevel Optimization Problem

On the Partial Calmness Condition for an Interval-Valued Bilevel Optimization Problem

Applying tangential subdifferentials in bilevel optimization

The problem considered in this paper is a sequence of two optimization problems in which the feasible region of the upper-level optimization problem is determined implicitly by the solution set of the lower-level optimization problem. Using the optimal value reformulation, together with the partial calmness property, we give necessary optimality conditions in terms of tangential subdifferentials. An example that illustrates our finding is also given.

Levenberg–Marquardt method and partial exact penalty parameter selection in bilevel optimization

We consider the optimistic bilevel optimization problem, known to have a wide range of applications in engineering, that we transform into a single-level optimization problem by means of the lower-level optimal value function reformulation. Subsequently, based on the partial calmness concept, we build an equation system, which is parameterized by the corresponding partial exact penalization parameter. We then design and analyze a Levenberg–Marquardt method to solve this parametric system of equations. Considering the fact that the selection of the partial exact penalization parameter is a critical issue when numerically solving a bilevel optimization problem by means of the value function reformulation, we conduct a careful experimental study to this effect, in the context of the Levenberg–Marquardt method, while using the Bilevel Optimization LIBrary (BOLIB) series of test problems. This study enables the construction of some safeguarding mechanisms for practical robust convergence of the method and can also serve as base for the selection of the penalty parameter for other bilevel optimization algorithms. We also compare the Levenberg–Marquardt method introduced in this paper to other existing algorithms of similar nature.

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.

R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

Some new optimality conditions for semivector bilevel optimization program

This paper discusses a kind of non-convex, non-smooth optimistic semivector bilevel optimization programs which equipped with a vector lower level problem. We introduce a class of new gap functions and penalty functions to transform this two level program into a one level scalar-objective optimization problem. Furthermore, we derive first-order optimality conditions for this semivector bilevel program in both global and local sense, using calculus of basic subdifferential and partial calmness. By applying new developments in basic subdifferential, we estimate basic subdifferential of gap functions and penalty functions which specify the former mentioned optimality conditions.

On Existence of Solutions of Parametrized Generalized Equations

In this paper we study existence of solutions to the generalized equation 0 ∈ f(p,x) + F(x), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F, local convex-valuedness of F− 1, and partial calmness of f with respect to x uniformly in p, for the property that, for any p near the reference value, the generalized equation has a solution at a certain distance from the reference solution. Some corollaries and applications of this result are also presented.

A note on partial calmness for bilevel optimization problems with linearly structured lower level

Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush–Kuhn–Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying issues with the aid of illustrative counterexamples.

Necessary optimality conditions for a semivectorial bilevel problem under a partial calmness condition

ABSTRACT In this paper, we are concerned with a semivectorial bilevel optimization problem Using a partial calmness suitable for bilevel semivectorial problems, we formulate its necessary optimality conditions. Our approach consists of reformulating our problem into a one level optimization problem using successively the kth-objective weighted-constraint and the optimal value reformulation. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first-order necessary optimality conditions are then derived in the smooth setting while using the generalized differentiation calculus of Mordukhovich.

ON THE PROBLEMS OF BILEVEL OPTIMIZATION UNDER RCPLD CONSTRAINT QUALIFICATIONS

Multilevel optimization problems often arise in various applications (in economics, ecology, power engineering and other areas) when modeling complex systems with a hierarchical structure associated with independent actions of subsystems. The difficulty of analyzing such complex systems requires first of all the study of bilevel models, the management of which would be an integral part of the analysis of more complex systems. In solving bilevel programming problems, an important role is played by the property of partial calmness, the presence of which allows us to reduce the bilevel problem to the classical nonlinear programming problem with a nonsmooth objective function. It is known that linear bilevel programming problems are partially stable. The proof of this property for more complex problems meets difficulties. In particular, our article shows the inaccuracy of some results in this area. The goal of the paper is to obtain some new results in the partial calmness of bilevel programming. In particular, new sufficient conditions for bilevel problems are proved. The results are obtained on the base of Lipschitz-like properties for multivalued mappings. In the paper we propose new sufficient conditions for partial calmness which are based on some modification of the known constraint qualification RCPLD which have been proposed by the researches Andreani, Haeser, Schuverdt and Silva.

On partial calmness for bilevel programming problems with lower-level problem linear in lower-level variable

Communicated by Corresponding Member Valentine V. Gorokhovik In the paper the Lipschitz-like properties of solution mapping of lower-level problems for bilevel programs are studied and sufficient conditions for partial calmness are proved on the basis of these properties.

The bilevel programming problem: reformulations, constraint qualifications and optimality conditions

We consider the bilevel programming problem and its optimal value and KKT one level reformulations. The two reformulations are studied in a unified manner and compared in terms of optimal solutions, constraint qualifications and optimality conditions. We also show that any bilevel programming problem where the lower level problem is linear with respect to the lower level variable, is partially calm without any restrictive assumption. Finally, we consider the bilevel demand adjustment problem in transportation, and show how KKT type optimality conditions can be obtained under the partial calmness, using the differential calculus of Mordukhovich.

On the Karush–Kuhn–Tucker reformulation of the bilevel optimization problem

This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M - and S -type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M -stationarity conditions.

Optimality results for a specific bilevel optimization problem

We develop necessary optimality conditions using techniques from variational analysis for a bilevel optimization problem without assuming the lower-level problem satisfying neither the Mangasarian Fromovitz constraint qualification nor the partial calmness condition [D.L. Ye and D.L. Zhu, Optimality conditions for bilevel programming problems, Optimization 33 (1995), pp. 9–27; J.J. Ye and J.J. Zhu, A note on optimality conditions for bilevel programming problems, Optimization 39 (1997), pp. 361–366]. Reducing the problem into a one-level nonlinear and nonsmooth programme, we use the approximate extremal principle [B.S. Mordukhovich, Variational Analysis and Generalized Differentiation: I. Basic Theory, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 330, Springer, Berlin, 2006; B.S. Mordukhovich, Variational Analysis and Generalized Differentiation: II. Applications, Grundlehren Series, Fundamental Principles of Mathematical Sciences, Vol. 331, Springer, Berlin, 2006] to get fuzzy and exact optimality conditions.

On calmness conditions in convex bilevel programming

In this article, we compare two different calmness conditions which are widely used in the literature on bilevel programming and on mathematical programs with equilibrium constraints. In order to do so, we consider convex bilevel programming as a kind of intersection between both research areas. The so-called partial calmness concept is based on the function value approach for describing the lower level solution set. Alternatively, calmness in the sense of multifunctions may be considered for perturbations of the generalized equation representing the same lower level solution set. Both concepts allow to derive first-order necessary optimality conditions via tools of generalized differentiation introduced by Mordukhovich. They are very different, however, concerning their range of applicability and the form of optimality conditions obtained. The results of this article seem to suggest that partial calmness is considerably more restrictive than calmness of the perturbed generalized equation. This fact is also illustrated by means of a dicretized obstacle control problem.

The Generalized Mangasarian-Fromowitz Constraint Qualification and Optimality Conditions for Bilevel Programs

We consider the optimal value reformulation of the bilevel programming problem. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of the basic generalized differentiation constructions of Mordukhovich, which is weaker than the one in terms of Clarke’s nonsmooth tools, fails without any restrictive assumption. Some weakened forms of this constraint qualification are then suggested, in order to derive Karush-Kuhn-Tucker type optimality conditions for the aforementioned problem. Considering the partial calmness, a new characterization is suggested and the link with the previous constraint qualifications is analyzed.

New Necessary Optimality Conditions for Bilevel Programs by Combining the MPEC and Value Function Approaches

The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush-Kuhn-Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.

A penalty function method based on bilevel programming for solving inverse optimal value problems

In this work, we reformulate the inverse optimal value problem equivalently as a corresponding nonlinear bilevel programming (BLP) problem. For the nonlinear BLP problem, the duality gap of the lower level problem is appended to the upper level objective with a penalty, and then a penalized problem is obtained. On the basis of the concept of partial calmness, we prove that the penalty function is exact. Then, an algorithm is proposed and an inverse optimal value problem is resolved to illustrate the algorithm.

Optimality Conditions for Vector Optimization Problems

In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem is proved under the assumption of partial calmness.