Solving bilevel programs with the KKT-approach

Allende, Gemayqzel Bouza; Still, Georg J.

doi:10.1007/s10107-012-0535-x

Cited by 106 publications

(63 citation statements)

References 20 publications

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“…Example 4.1 in Dempe and Dutta [25] can be used to show that the (MFCQ) is not a constraint qualification for the lower level problem which is generically satisfied. Knowing this, Allende and Still suggest in [27] to replace the lower level problem (1.2) not by its Karush-Kuhn-Tucker optimality conditions but by the Fritz-John necessary optimality conditions since the Mangasarian-Fromovitz constraint qualification for the lower level problem does not need to be satisfied at an optimal solution of the bilevel problem. The resulting problem reads as…”

Section: Introductionmentioning

confidence: 96%

“…It is shown in this paper [27] that for almost all linear perturbations of the functions F, G, f, g, a critical point (x, y, λ) of (1.4) is either an isolated non-degenerate critical point of this problem or the lower level problem partially vanishes. In the last case, the point (x, y) is a critical point of minimizing the upper level objective function subject to the upper and lower level constraints, and the multiplier λ is a singular multiplier of the lower level problem.…”

Section: Introductionmentioning

confidence: 97%

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On the solution of convex bilevel optimization problems

Dempe

Franke

2015

Comput Optim Appl

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An algorithm is presented for solving bilevel optimization problems with fully convex lower level problems. Convergence to a local optimal solution is shown under certain weak assumptions. This algorithm uses the optimal value transformation of the problem. Transformation of the bilevel optimization problem using the FritzJohn necessary optimality conditions applied to the lower level problem is shown to exhibit almost the same difficulties for solving the problem as the use of the KarushKuhn-Tucker conditions.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

On the solution of convex bilevel optimization problems

Dempe

Franke

2015

Comput Optim Appl

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“…Basic definitions and concepts from variational analysis. For a closed subset C of R n , the basic (or limiting/Mordukhovich) normal cone to C at one of its pointsx is the set 1) where N C denotes the dual of the contingent/Bouligand tangent cone to C. Note that if C := ψ −1 (Ξ), where Ξ ⊆ R m is a closed set and ψ [R n → R m ] a Lipschitz continuous function aroundx, then we have…”

mentioning

confidence: 99%

“…Moreover, the linear independence constraint qualification is not a generic regularity condition in the lower-level problem [4], at least in the case when the lowerlevel constraints depend on the parameter. In [1], the more general problem where the KKT conditions of the lower-level problem are replaced by the Fritz-John conditions is considered and the generic structure of the feasible region is studied. It is important to mention that the resulting problem extremely modifies the bilevel optimization problem.…”

mentioning

confidence: 99%

KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

Dempe¹,

Zemkoho²

2014

SIAM J. Optim.

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Abstract. For 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.

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“…In the first strategy the bi-level structure of the problem is kept and exploited [25,26,29]. In the second one the bi-level problem is replaced by a one-level problem, substituting the lower level problem by its optimality conditions [8,9,14,5,13,2]. In our research we consider and also compare both approaches.…”

Section: B Solution Strategymentioning

confidence: 99%

An Inverse Optimal Control Approach for the Transfer of Human Walking Motions in Constrained Environment to Humanoid Robots

Clever¹,

Mombaur²

Robotics: Science and Systems XII

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Abstract-In this paper we present an inverse optimal control based transfer of motions from human experiments to humanoid robots and apply it to walking in constrained environments. To this end we introduce a 3D template model, which describes motion on the basis of center of mass trajectory, foot trajectories, upper body orientation and phase duration. Despite of its abstract architecture with prismatic joints combined with damped series elastic actuators instead of knees, the model (including dynamics and constraints) is suitable to describe both, human and humanoid locomotion with appropriate parameters. We present and apply an inverse optimal control approach to identify optimality criteria based on human motion capture experiments. The identified optimal strategy is then transferred to the humanoid robot for gait generation by solving an optimal control problem, which takes into account the properties of the robot and differences in the environment. The results of this approach are the center of mass trajectory, the foot trajectories, the torso orientation, and the single and double support phase durations for a sequence of multiple steps allowing the humanoid robot to walk within a new environment. We present one computational cycle (from motion capture data to an optimized robot motion) for the example of walking over irregular step stones with the aim to transfer the motion to two very different humanoid robots (iCub Heidelberg01 and HRP-2 14). The transfer of these optimized robot motions to the real robots by means of inverse kinematics is work in progress and not part of this paper.

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Solving bilevel programs with the KKT-approach

Cited by 106 publications

References 20 publications

On the solution of convex bilevel optimization problems

On the solution of convex bilevel optimization problems

KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

An Inverse Optimal Control Approach for the Transfer of Human Walking Motions in Constrained Environment to Humanoid Robots

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