Dominik Dorsch

ON STRUCTURE AND COMPUTATION OF GENERALIZED NASH EQUILIBRIA

We consider generalized Nash equilibrium problems (GNEP) from a structural and computational point of view. In GNEP the players’ feasible sets may depend on the other players’ strategies. Moreover, the players may share common constraints. In particular, the latter leads to the stable appearance of Nash equilibria which are Fritz-John (FJ) points, but not Karush-Kuhn-Tucker (KKT) points. Basic in our approach is the representation of FJ points as zeros of an appropriate underdetermined system of nonsmooth equations. Here, additional nonsmooth variables are taken into account. We prove that the set of FJ points (together with corresponding active Lagrange multipliers) - generically - constitutes a Lipschitz manifold. Its dimension is (N-1)J_0, where N is the number of players and J_0 is the number of active common constraints. In a structural analysis of Nash equilibria the number (N-1)J_0 plays a crucial role. In fact, the latter number encodes both the possible degeneracies for the players’ parametric subproblems and the dimension of the set of Nash equilibria. In particular, in the nondegenerate case, the dimension of the set of Nash equilibria equals locally (N-1)J_0. For the computation of FJ points we propose a nonsmooth projection method (NPM) which aims at nding solutions of an underdetermined system of nonsmooth equations. NPM is shown to be well-dened for GNEP. Local convergence of NPM is conjectured for GNEP under generic assumptions and its proof is challenging. However, we indicate special cases (known from the literature) in which convergence holds.

Mathematical programs with vanishing constraints: critical point theory

We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C2-perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.

Refined analysis of sparse MIMO radar

We analyze a multiple-input multiple-output (MIMO) radar model and provide recovery results for a compressed sensing (CS) approach. In MIMO radar different pulses are emitted by several transmitters and the echoes are recorded at several receiver nodes. Under reasonable assumptions the transformation from emitted pulses to the received echoes can approximately be regarded as linear. For the considered model, and many radar tasks in general, sparsity of targets within the considered angle-range-Doppler domain is a natural assumption. Therefore, it is possible to apply methods from CS in order to reconstruct the parameters of the targets. Assuming Gaussian random pulses the resulting measurement matrix becomes a highly structured random matrix. Our first main result provides an estimate for the well-known restricted isometry property (RIP) ensuring stable and robust recovery. We require more measurements than standard results from CS, like for example those for Gaussian random measurements. Nevertheless, we show that due to the special structure of the considered measurement matrix our RIP result is in fact optimal (up to possibly logarithmic factors). Our further two main results on nonuniform recovery (i.e., for a fixed sparse target scene) reveal how the fine structure of the support set—not only the size—affects the (nonuniform) recovery performance. We show that for certain “balanced” support sets reconstruction with essentially the optimal number of measurements is possible. Indeed, we introduce a parameter measuring the well-behavedness of the support set and resemble standard results from CS for near-optimal parameter choices. We prove recovery results for both perfect recovery of the support set in case of exactly sparse vectors and an

Analysis of sparse recovery in MIMO radar

\ell _2

On implicit functions in nonsmooth analysis

ℓ2-norm approximation result for reconstruction under sparsity defect. Our analysis complements earlier work by Strohmer & Friedlander and deepens the understanding of the considered MIMO radar model. Thereby—and apparently for the first time in CS theory—we prove theoretical results in which the difference between nonuniform and uniform recovery consists of more than just logarithmic factors.

On Intrinsic Complexity of Nash Equilibrium Problems and Bilevel Optimization

We derive a new genericity result for nonlinear semidefinite programming (NLSDP). Namely, almost all linear perturbations of a given NLSDP are shown to be nondegenerate. Here, nondegeneracy for NLSDP refers to the transversality constraint qualification, strict complementarity and second-order sufficient condition. Due to the presence of the second-order sufficient condition, our result is a nontrivial extension of the corresponding results for linear semidefinite programs (SDP) from Alizadeh et al. (Math Program 77(2, Ser. B):111–128, 1997). The proof of the genericity result makes use of Forsgren’s derivation of optimality conditions for NLSDP in Forsgren (Math Program 88(1, Ser. A):105–128, 2000). Due to the latter approach, the positive semidefiniteness of a symmetric matrix G(x), depending continuously on x, is locally equivalent to the fact that a certain Schur complement S(x) of G(x) is positive semidefinite. This yields a reduced NLSDP by considering the new semidefinite constraint

MPVC: Critical Point Theory

S(x) \succeq 0