Results 1 
3 of
3
On Minimal Valid Inequalities for Mixed Integer Conic Programs
"... 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 Kminimal inequalities and show that, under mild assumptions, ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
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 Kminimal inequalities and show that, under mild assumptions, these inequalities together with the trivial coneimplied inequalities are sufficient to describe the convex hull. We focus on the properties ofKminimal inequalities by establishing algebraic necessary conditions for an inequality to be Kminimal. This characterization leads to a broader algebraically defined class of Ksublinear inequalities. We demonstrate a close connection between Ksublinear inequalities and the support functions of convex sets with a particular structure. This connection results in practical ways of verifying Ksublinearity and/or Kminimality of inequalities. Our study 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 which are also piecewise linear. Our analysis easily recovers this result. However, in the case of general regular cones other than the nonnegative orthant, our study reveals that such a cutgenerating function view that treats the data associated with each individual variable independently is far from sufficient.
CBLIB 2014: A benchmark library for conic mixedinteger and continuous optimization. Optimization Online
, 2014
"... Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already suppor ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already supported by all major commercial solvers and spans a wide range of industrial applications. The library addresses the particular need for public test sets mixing cone types and allowing integer variables, but has all types of conic optimization in target. The CBF file format is embraced as standard, and tools are provided to aid integration with, or transformation to the input format of, any software package.
Noname manuscript No. (will be inserted by the editor) On Sublinear Inequalities for Mixed Integer Conic Programs
, 2014
"... Abstract This paper studies Ksublinear inequalities, a class of inequalities with strong relations to Kminimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of Ksublinear inequalities. That is, we show that when K is the nonnegative orthant or the seco ..."
Abstract
 Add to MetaCart
Abstract This paper studies Ksublinear inequalities, a class of inequalities with strong relations to Kminimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of Ksublinear inequalities. That is, we show that when K is the nonnegative orthant or the secondorder cone, Ksublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When K is the nonnegative orthant, Ksublinear inequalities are tightly connected to functions that generate cuts—so called cutgenerating functions. As a consequence of the sufficiency of Rn+sublinear inequalities, we also provide an alternate and straightforward proof of the sufficiency of cutgenerating functions for mixed integer linear programs, a result recently established by Cornuéjols, Wolsey and Yıldız. 1