Solving semi-infinite programs by smoothing projected gradient method

Solving semi-infinite programs by smoothing projected gradient method

In this paper, we study a semi-infinite programming (SIP) problem with a convex set constraint. Using the value function of the lower level problem, we reformulate SIP problem as a nonsmooth optimization problem. Using the theory of nonsmooth Lagrange multiplier rules and Danskin’s theorem, we present constraint qualifications and necessary optimality conditions. We propose a new numerical meth...

A Smoothing Newton Method for Semi-Infinite Programming

Solving Semi-infinite Linear Programs Using Boosting-Like Methods

Linear optimization problems (LPs) with a very large or even infinite number of constraints frequently appear in many forms in machine learning. A linear program with m constraints can be written as min x∈P n c x with a where I assume for simplicity that the domain of x is the n dimensional probability simplex P n. Optimization problems with an infinite number of constraints of the form a j x ≤...

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a...

Smoothing by mollifiers. Part I: semi-infinite optimization

We show that a compact feasible set of a standard semi-infinite optimization problem can be approximated arbitrarily well by a level set of a single smooth function with certain regularity properties. This function is constructed as the mollification of the lower level optimal value function. Moreover, we use correspondences between KarushKuhn-Tucker points of the original and the smoothed prob...