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Exact augmented Lagrangian duality for mixed integer convex optimization

Avinash Bhardwaj, Vishnu Narayanan, Abhishek Pathapati

TL;DR

This work presents a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions.

Abstract

Augmented Lagrangian dual augments the classical Lagrangian dual with a non-negative non-linear penalty function of the violation of the relaxed/dualized constraints in order to reduce the duality gap. We investigate the cases in which mixed integer convex optimization problems have an exact penalty representation using sharp augmenting functions (norms as augmenting penalty functions). We present a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions. This generalizes the recent results for MILP (Feizollahi, Ahmed and Sun, 2017) and MIQP (Gu, Ahmed and Dey 2020) whilst also providing an alternative proof for the aforementioned along with quantification of the finite penalty parameter in these cases.

Exact augmented Lagrangian duality for mixed integer convex optimization

TL;DR

This work presents a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions.

Abstract

Augmented Lagrangian dual augments the classical Lagrangian dual with a non-negative non-linear penalty function of the violation of the relaxed/dualized constraints in order to reduce the duality gap. We investigate the cases in which mixed integer convex optimization problems have an exact penalty representation using sharp augmenting functions (norms as augmenting penalty functions). We present a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions. This generalizes the recent results for MILP (Feizollahi, Ahmed and Sun, 2017) and MIQP (Gu, Ahmed and Dey 2020) whilst also providing an alternative proof for the aforementioned along with quantification of the finite penalty parameter in these cases.
Paper Structure (11 sections, 23 theorems, 130 equations)

This paper contains 11 sections, 23 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Main results
  5. Proofs of the Theorems
  6. Preliminary Lemmas
  7. Proof of \ref{['thm:1']}
  8. Proof of \ref{['thm:2']}
  9. Proof of \ref{['thm:3']}
  10. MICPs with implicit integer boundedness
  11. A Special Case: Pure Integer Convex Programs

Key Result

Theorem 3.1

\newlabelthm:110 Consider the following mixed integer linear programming problem, There exists an exact penalty representation for (MILP). Furthermore, the finite penalty parameter $\rho$ depends on $\mathbf{A}_C$ and $\mathbf{c}_C$ and $\mathbf{c}_I$ .

Theorems & Definitions (43)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Lemma 4.1
  • Proof 1
  • Proposition 4.2
  • Proof 2
  • Corollary 4.3
  • Proof 3
  • ...and 33 more