Fatma Kılınç-Karzan

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-script) such as the nonnegative orthant, the Lorentz cone, or the positive semidefinite cone. In a unified framework, we introduce (K-script) -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 focus on the properties of (K-script) -minimal inequalities by establishing algebraic necessary conditions for an inequality to be (K-script) -minimal. This characterization leads to a broader algebraically defined class of (K-script) -sublinear inequalities. We demonstrate a close connection between (K-script) -sublinear inequalities and the support functions of convex sets with a particular structure. This connection results in practical ways of verifying (K-script) -sublinearity and/or (K-script) -minimality 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 that 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 cut-generating function view, which treats the data associated with each individual variable independently, is far from sufficient.

Two-Term Disjunctions on the Second-Order Cone

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we develop a methodology to derive closed-form expressions for inequalities describing the convex hull of a two-term disjunction applied to the second-order cone. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly nonlinear, valid inequalities that correspond to these linear inequalities. We identify and study the cases where these valid inequalities can equivalently be expressed in conic quadratic form and where a single inequality from this family is sufficient to describe the convex hull. Our results on two-term disjunctions on the second-order cone generalize related results on split cuts by Modaresi, Kilinc, and Vielma, and by Andersen and Jensen.

How to convexify the intersection of a second order cone and a nonconvex quadratic

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid,

Approximating the stability region for binary mixed-integer programs

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Faster First-Order Methods for Extensive-Form Game Solving

E, and a split disjunction,

Business Analytics Assists Transitioning Traditional Medicine to Telemedicine at Virtual Radiologic

(l - x_j)(x_j - u) \le 0