124,618 research outputs found
Solving ill-posed bilevel programs
No full textThis paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem
KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization
No full textFor a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper
The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs
No full textWe 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
A simple approach to optimality conditions in minmax programing
No full textConsidering the minmax programming problem, lower and upper subdifferential optimality conditions, in the sense of Mordukhovich, are derived. The approach here, mainly based on the nonsmooth dual objects of Mordukhovich, is completely different from that of most of the previous works where generalizations of the alternative theorem of Farkas have been applied. The results obtained are close to those known in the literature. However, one of the main achievements of this article is that we could also derive necessary optimality conditions for the minmax program of the usual Karush–Kuhn–Tucker type, which seems to be new in this field of study
On the Karush–Kuhn–Tucker reformulation of the bilevel optimization problem
No full textThis 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
Bilevel road pricing: theoretical analysis and optimality conditions
No full textWe consider the bilevel road pricing problem. In contrary to the Karush-Kuhn-Tucker (one level) reformulation, the optimal value reformulation is globally and locally equivalent to the initial problem. Moreover, in the process of deriving optimality conditions, the optimal value reformulation helps to preserve some essential data involved in the traffic assignment problem that may disappear with the Karush-Kuhn-Tucker (KKT) one. Hence, we consider in this work the optimal value reformulation of the bilevel road pricing problem; using some recent developments in nonsmooth analysis, we derive implementable KKT type optimality conditions for the problem containing all the necessary information. The issue of estimating the (fixed) demand required for the road pricing problem is a quite difficult problem which has been also addressed in recent years using bilevel programming. We also show how the ideas used in designing KKT type optimality conditions for the road pricing problem can be applied to derive optimality conditions for the origin-destination (O-D) matrix estimation problem. Many other theoretical aspects of the bilevel road pricing and O-D matrix estimation problems are also studied in this paper
The bilevel programming problem: reformulations, constraint qualifications and optimality conditions
No full textWe 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
Levenberg-Marquardt method and partial exact penalty parameter selection in bilevel optimization
No full textWe 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, we conduct a careful experimental study to this effect, in the context the Levenberg-Marquardt method, while using the Bilevel Optimization LIBrary (BOLIB) series of test problems
New optimality conditions for the semivectorial bilevel optimization problem
No full textThe paper is concerned with the optimistic formulation of a bilevel optimization problem with multiobjective lower-level problem. Considering the scalarization approach for the multiobjective program, we transform our problem into a scalar-objective optimization problem with inequality constraints by means of the well-known optimal value reformulation. Completely detailed first-order necessary optimality conditions are then derived in the smooth and nonsmooth settings while using the generalized differentiation calculus of Mordukhovich. Our approach is different from the one previously used in the literature and the conditions obtained are new. Furthermore, they reduce to those of a usual bilevel program, if the lower-level objective function becomes single-valued
Sensitivity analysis for two-level value functions with applications to bilevel programming
No full textThis paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization called the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not - to a large extent - differ from those known for the conventional problem. It has already been well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This is related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the two-level value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data
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