[Paper Review] On Minimal Valid Inequalities for Mixed Integer Conic Programs

Fatma Kılınç-Karzan|arXiv (Cornell University)|Aug 29, 2014

Advanced Optimization Algorithms Research46 references4 citations

TL;DR

This paper introduces a unified framework for minimal valid inequalities in mixed integer conic programs (MICPs) using regular cones such as the nonnegative orthant, Lorentz cone, and positive semidefinite cone. It defines K-minimal inequalities and shows they, along with cone-implied inequalities, describe the convex hull under mild assumptions, generalizing results from mixed integer linear programming and revealing new algebraic and geometric characterizations via support functions.

ABSTRACT

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone.

Motivation & Objective

• To develop a unified framework for minimal valid inequalities in mixed integer conic programs involving general regular cones.
• To characterize K-minimal inequalities algebraically and relate them to sublinear functions and support functions.
• To generalize known results from mixed integer linear programming to the conic case, particularly the role of piecewise linear sublinear functions.
• To investigate the structure and finiteness of K-minimal inequalities beyond the nonnegative orthant, especially for Lorentz and semidefinite cones.
• To provide practical conditions for verifying K-minimality and K-sublinearity of valid inequalities in MICPs.

Proposed method

• Introduces the concept of K-minimal inequalities for disjunctive conic sets defined by a regular cone K and a finite set of constraints.
• Defines K-sublinear inequalities as a broader class that includes K-minimal ones, characterized by positive homogeneity, subadditivity, and convexity.
• Establishes a connection between K-sublinear inequalities and support functions of specific convex sets, enabling practical verification.
• Uses this support function link to derive sufficient conditions for K-minimality and K-sublinearity, even in non-polyhedral cones.
• Extends the notion of K-minimality to conic inequalities by requiring all associated linear inequalities to be K-minimal.
• Applies the framework to recover classical MILP results, showing that K-sublinear functions reduce to piecewise linear sublinear functions when K is the nonnegative orthant.

Experimental results

Research questions

• RQ1 Can K-minimal inequalities be characterized algebraically for general regular cones beyond the nonnegative orthant?
• RQ2 What is the relationship between K-sublinear inequalities and support functions of structured convex sets?
• RQ3 Are there practical sufficient conditions to verify K-minimality or K-sublinearity of a given inequality in MICPs?
• RQ4 Is the number of K-minimal conic inequalities required to describe the convex hull of a disjunctive conic set finite when the underlying cone is non-polyhedral?
• RQ5 Can the cut-generating function approach, effective in MILPs, be generalized to MICPs with non-orthant cones like the Lorentz cone?

Key findings

• K-minimal inequalities, together with cone-implied inequalities, are sufficient to describe the convex hull of a disjunctive conic set under mild assumptions.
• When K is the nonnegative orthant, all K-sublinear inequalities are generated by piecewise linear, positively homogeneous, subadditive, and convex functions, recovering classical MILP results.
• A strong connection is established between K-sublinear inequalities and support functions of specific convex sets, enabling practical verification of K-minimality.
• For non-polyhedral cones such as the Lorentz cone L^3, the convex hull may require infinitely many extreme linear inequalities, but only finitely many conic inequalities of the same type.
• The framework reveals that treating data per variable independently, as in MILP cut-generating functions, is not generally possible for cones other than the nonnegative orthant, even for the Lorentz cone.
• The paper provides new sufficient conditions for K-minimality and K-sublinearity, which are novel even in the MILP setting.