Convex semi-infinite programming algorithms with inexact separation oracles

TL;DR

This paper develops algorithms for convex semi-infinite programming that effectively handle inexact separation oracles, providing convergence guarantees and feasibility preservation, especially for strongly convex objectives and quadratic constraints.

Contribution

It introduces and analyzes the convergence of CP and IOA algorithms for SIP with inexact separation oracles, including specific results for quadratic constraints.

Findings

CP algorithm converges at rate O(1/k) with strong convexity

IOA algorithm maintains feasibility despite oracle inaccuracy

Convergence proven for quadratic constrained SIPs with inexact oracles

Abstract

Solving convex Semi-Infinite Programming (SIP) problems is challenging when the separation problem, i.e., the problem of finding the most violated constraint, is computationally hard. We propose to tackle this difficulty by solving the separation problem approximately, i.e., by using an inexact oracle. Our focus lies in two algorithms for SIP, namely the Cutting-Planes (CP) and the Inner-Outer Approximation (IOA) algorithms. We prove the CP convergence rate to be in O(1/k), where k is the number of calls to the limited-accuracy oracle, if the objective function is strongly convex. Compared to the CP algorithm, the advantage of the IOA algorithm is the feasibility of its iterates. In the case of a semi-infinite program with Quadratically Constrained Quadratic Programming separation problem, we prove the convergence of the IOA algorithm toward an optimal solution of the SIP problem despite the oracle's inexactness.