Alexandra Schwartz

Mathematical Programs with Cardinality Constraints: Reformulation by Complementarity-Type Conditions and a Regularization Method

Optimization problems with cardinality constraints are very difficult mathematical programs which are typically solved by global techniques from discrete optimization. Here we introduce a mixed-integer formulation whose standard relaxation still has the same solutions (in the sense of global minima) as the underlying cardinality-constrained problem; the relation between the local minima is also discussed in detail. Since our reformulation is a minimization problem in continuous variables, it allows us to apply ideas from that field to cardinality-constrained problems. Here, in particular, we therefore also derive suitable stationarity conditions and suggest an appropriate regularization method for the solution of optimization problems with cardinality constraints. This regularization method is shown to be globally convergent to a Mordukhovich-stationary point. Extensive numerical results are given to illustrate the behavior of this method.

The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, 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 prominent class of specialized algorithms is the relaxation (or regularization) methods. The first relaxation method for MPECs is due to Scholtes [35] [Scholtes S (2001) Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM Journal on Optimization 11:918–936.], but in the meantime, there exists a number of different regularization schemes that try to relax the difficult constraints in different ways. Some of these more recent schemes have better theoretical properties than does the original method by Scholtes. Nevertheless, numerical experience shows that the Scholtes relaxation method is still among the fastest and most...

Constraint qualifications and optimality conditions for optimization problems with cardinality constraints

This paper considers optimization problems with cardinality constraints. Based on a recently introduced reformulation of this problem as a nonlinear program with continuous variables, we first define some problem-tailored constraint qualifications and then show how these constraint qualifications can be used to obtain suitable optimality conditions for cardinality constrained problems. Here, the (KKT-like) optimality conditions hold under much weaker assumptions than the corresponding result that is known for the somewhat related class of mathematical programs with complementarity constraints.

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

Numerische Verfahren für VIPs

In diesem Kapitel beschreiben wir verschiedene Verfahren zur Losung der Variationsungleichung (mathrm{VIP}(X,F)): Fur gegebenes (Xsubseteqmathbb{R}^{n}) und (Fcolon Xtomathbb{R}^{n}) finde (x^{*}in X) mit n n

Second-Order Optimality Conditions and Improved Convergence Results for Regularization Methods for Cardinality-Constrained Optimization Problems

displaystyle F(x^{*})^{T}(x-x^{*})geq 0quadforall xin X.

Lottery versus All-Pay Auction Contests – A Revenue Dominance Theorem

n nIm Hinblick auf die Satze 3.5 und 5.6 konnen diese Verfahren insbesondere auch zur Losung von (verallgemeinerten) Nash-Gleichgewichtsproblemen benutzt werden. Dies ist in der Tat ein gangiger Weg zur numerischen Losung von (normalisierten) Nash-Gleichgewichtsproblemen, denn die Verfahren fur Variationsungleichungen sind mittlerweile sehr ausgereift, wahrend es vergleichsweise weniger Verfahren zur Losung von NEPs und GNEPs gibt.

A Generalized Nash Game for Mobile Edge Computation Offloading

Bisher haben wir Spiele der Form n n