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Polynomial Optimization Relaxations for Generalized Semi-Infinite Programs

Xiaomeng Hu, Jiawang Nie

TL;DR

This work addresses solving generalized semi-infinite programs with polynomial data by introducing a hierarchy of polynomial optimization relaxations that hinge on polynomial extensions of x-dependent infinity sets. The core approach combines Lagrange multiplier expressions with Moment-SOS relaxations to handle the infinite constraints, achieving either finite or asymptotic convergence under regularity conditions. Key contributions include explicit construction of polynomial extensions for common feasible sets, an efficient algorithm that updates relaxations via these extensions, and a convex-infinity constraint specialization that can yield exact solutions from a single relaxation. The methods are validated through extensive numerical experiments on SIPs and GSIPs, demonstrating practical efficiency and broad applicability to problems arising in design, optimization, and control. The framework offers a promising, scalable path for global optimization in polynomial GSIPs with strong certificate capabilities via SOS techniques.

Abstract

This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions. Moment-SOS relaxations are applied to solve the polynomial optimization. The convergence of this hierarchy is shown under certain conditions. In particular, the classical semi-infinite programs (SIPs) can be solved as a special case of GSIPs. We also study GSIPs that have convex infinity constraints and show that they can be solved exactly by a single polynomial optimization relaxation. The computational efficiency is demonstrated by extensive numerical results.

Polynomial Optimization Relaxations for Generalized Semi-Infinite Programs

TL;DR

This work addresses solving generalized semi-infinite programs with polynomial data by introducing a hierarchy of polynomial optimization relaxations that hinge on polynomial extensions of x-dependent infinity sets. The core approach combines Lagrange multiplier expressions with Moment-SOS relaxations to handle the infinite constraints, achieving either finite or asymptotic convergence under regularity conditions. Key contributions include explicit construction of polynomial extensions for common feasible sets, an efficient algorithm that updates relaxations via these extensions, and a convex-infinity constraint specialization that can yield exact solutions from a single relaxation. The methods are validated through extensive numerical experiments on SIPs and GSIPs, demonstrating practical efficiency and broad applicability to problems arising in design, optimization, and control. The framework offers a promising, scalable path for global optimization in polynomial GSIPs with strong certificate capabilities via SOS techniques.

Abstract

This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions. Moment-SOS relaxations are applied to solve the polynomial optimization. The convergence of this hierarchy is shown under certain conditions. In particular, the classical semi-infinite programs (SIPs) can be solved as a special case of GSIPs. We also study GSIPs that have convex infinity constraints and show that they can be solved exactly by a single polynomial optimization relaxation. The computational efficiency is demonstrated by extensive numerical results.
Paper Structure (18 sections, 4 theorems, 154 equations, 1 figure)

This paper contains 18 sections, 4 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Moment-SOS relaxations
  4. Polynomial optimization relaxations
  5. Polynomial extensions
  6. The algorithm
  7. Convergence properties
  8. Polynomial Extensions
  9. Convex Infinity Constraints
  10. Numerical Experiments
  11. SIP examples
  12. GSIP examples
  13. GSIPs with convex infinite constraints
  14. GSIPs from existing references
  15. Conclusions and Discussions
  16. ...and 3 more sections

Key Result

Theorem 3.4

In Algorithm algorithm, if $\hat{g}_k \ge 0$ at the $k$th loop, then the minimizer $\hat{x}_k$ for $(P_k)$ is also a minimizer for primal-GSIP.

Figures (1)

  • Figure 1: The shaded area is the region of $g(u) \ge 0$ in Example \ref{['four_application_examples']}. The dotted line is the boundary of the maximum ellipsoid inscribed in the shade.

Theorems & Definitions (38)

  • Example 3.1
  • Definition 3.2
  • Theorem 3.4: Finite convergence
  • proof
  • Theorem 3.5: Asymptotic convergence
  • proof
  • Example 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Example 6.1
  • ...and 28 more