Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method

Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, etc. Often, these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization one develops primal heuristics in order to obtain … Read more

The nonsmooth landscape of phase retrieval

We consider a popular nonsmooth formulation of the real phase retrieval problem. We show that under standard statistical assumptions, a simple subgradient method converges linearly when initialized within a constant relative distance of an optimal solution. Seeking to understand the distribution of the stationary points of the problem, we complete the paper by proving that … Read more

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

Optimization problems with cardinality constraints are very dicult 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 … Read more

Quantitative Stability Analysis of Stochastic Generalized Equations

We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random set-valued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the set-valued mapping with … Read more

A Bilevel Direct Search Method for Leader-Follower Optimization Problems and Applications

In the paper, we propose a bilevel direct search method for solving a type of leader-follower problems with each decision maker’s objective being a “black-box” function. First, we give a description for a leader-follower optimization problem. Then, we investigate a bilevel direct search method including two algorithms for combinatorially solving the upper and lower level … Read more