KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization

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

Optimality conditions for abs-normal NLPs

Structured nonsmoothness is widely present in practical optimization problems. A particularly attractive class of nonsmooth problems, both from a theoretical and from an algorithmic perspective, are nonsmooth NLPs with equality and inequality constraints in abs-normal form, so-called abs-normal NLPs. In this thesis optimality conditions for this particular class are obtained. To this aim, first the theory for the case of unconstrained optimization problems in abs-normal form of Andreas Griewank and Andrea Walther is extended. In particular, similar necessary and sufficient conditions of first and second order are obtained that are directly based on classical Karush-Kuhn-Tucker (KKT) theory for smooth NLPs. Then, it is shown that the class of abs-normal NLPs is equivalent to the class of Mathematical Programs with Equilibrium Constraints (MPECs). Hence, the regularity assumption LIKQ introduced for the abs-normal NLP turns out to be equivalent to MPEC-LICQ. Moreover, stationarity concepts and optimality conditions under these regularity assumptions of linear independece type are equivalent up to technical assumptions. Next, well established constraint qualifications of Mangasarian Fromovitz, Abadie and Guignard type for MPECs are used to define corresponding concepts for abs-normal NLPs. Then, it is shown that kink qualifications and MPEC constraint qualifications of Mangasarian Fromovitz resp. Abadie type are equivalent. As it remains open if this holds for Guignard type kink and constraint qualifications, branch formulations for abs-normal NLPs and MPECs are introduced. Then, equivalence of Abadie’s and Guignard’s constraint qualifications for all branch problems hold. Throughout a reformulation of inequalities with absolute value slacks is considered. It preserves constraint qualifications of linear independence and Abadie type but not of Mangasarian Fromovitz type. For Guignard type it is still an open question but ACQ and GCQ are preserved passing over to branch problems. Further, M-stationarity and B-stationarity concepts for abs-normal NLPs are introduced and corresponding first order optimality con- ditions are proven using the corresponding concepts for MPECs. Moreover, a reformulation to extend the optimality conditions for abs-normal NLPs to those with additional nonsmooth objective functions is given and the preservation of regularity assumptions is considered. Using this, it is shown that the unconstrained abs-normal NLP always satisfies constraint qualifications of Abadie and thus Guignard type. Hence, in this special case every local minimizer satisfies the M-stationarity and B-stationarity concepts for abs-normal NLPs

Using EPECs to model bilevel games in restructured electricity markets with locational prices

CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context

Validation of nominations in gas networks and properties of technical capacities

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A novel approach for bilevel programs based on Wolfe duality

This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be reformulated as a mathematical program with equilibrium constraints (MPEC for short). However, since the MPEC does not satisfy the Mangasarian-Fromovitz constraint qualification at any feasible point, the well-developed nonlinear programming theory cannot be applied to MPECs directly. In this paper, we apply the Wolfe duality to show that, under very mild conditions, the bilevel program is equivalent to a new single-level reformulation (WDP for short) in the globally and locally optimal sense. We give an example to show that, unlike the MPEC reformulation, WDP may satisfy the Mangasarian-Fromovitz constraint qualification at its feasible points. We give some properties of the WDP reformulation and the relations between the WDP and MPEC reformulations. We further propose a relaxation method for solving WDP and investigate its limiting behavior. Comprehensive numerical experiments indicate that, although solving WDP directly does not perform very well in our tests, the relaxation method based on the WDP reformulation is quite efficient

Solving Mathematical Programs with Equilibrium Constraints as Nonlinear Programming: A New Framework

We present a new framework for the solution of mathematical programs with equilibrium constraints (MPECs). In this algorithmic framework, an MPECs is viewed as a concentration of an unconstrained optimization which minimizes the complementarity measure and a nonlinear programming with general constraints. A strategy generalizing ideas of Byrd-Omojokun's trust region method is used to compute steps. By penalizing the tangential constraints into the objective function, we circumvent the problem of not satisfying MFCQ. A trust-funnel-like strategy is used to balance the improvements on feasibility and optimality. We show that, under MPEC-MFCQ, if the algorithm does not terminate in finite steps, then at least one accumulation point of the iterates sequence is an S-stationary point