Cutting Plane Methods and Subgradient Methods

Interior point methods have proven very successful at solving linear programming problems. When an explicit linear programming formulation is either not available or is too large to employ directly, a column generation approach can be used. Examples of column generation approaches include cutting plane methods for integer programming and decomposition methods for many classes of … Read more

An LPCC Approach to Nonconvex Quadratic Programs

Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving a linear program with linear complementarity constraints, i.e., an LPCC. Alternatively, this task can be … Read more

Properties of a Cutting Plane Method for Semidefinite Programming

We analyze the properties of an interior point cutting plane algorithm that is based on a semi-infinite linear formulation of the dual semidefinite program. The cutting plane algorithm approximately solves a linear relaxation of the dual semidefinite program in every iteration and relies on a separation oracle that returns linear cutting planes. We show that … Read more

On the Global Solution of Linear Programs with Linear Complementarity Constraints

This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of … Read more

Selective Gram-Schmidt orthonormalization for conic cutting surface algorithms

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly … Read more

An analytic center cutting plane approach for conic programming

We analyze the problem of finding a point strictly interior to a bounded, fully dimensional set from a finite dimensional Hilbert space. We generalize the results obtained for the LP, SDP and SOCP cases. The cuts added by our algorithm are central and conic. In our analysis, we find an upper bound for the number … Read more

A second-order cone cutting surface method: complexity and application

We present an analytic center cutting surface algorithm that uses mixed linear and multiple second-order cone cuts. Theoretical issues and applications of this technique are discussed. From the theoretical viewpoint, we derive two complexity results. We show that an approximate analytic center can be recovered after simultaneously adding $p$ second-order cone cuts in $O(p\log(p+1))$ Newton … Read more

Finding optimal realignments in sports leagues using a branch-and-cut-and-price approach

The sports team realignment problem can be modelled as $k$-way equipartition: given a complete graph $K_{n}=(V,E)$, with edge weight $c_{e}$ on each edge, partition the vertices $V$ into $k$ divisions that have exactly $S$ vertices, so as to minimize the total weight of the edges that have both endpoints in the same division. In this … Read more

Rebalancing an Investment Portfolio in the Presence of Convex Transaction Costs

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which convex transaction costs are incurred to rebalance an investment portfolio. In particular, we consider linear, piecewise linear, and quadratic transaction costs. The Markowitz framework of mean-variance efficiency … Read more