Gábor Pataki

On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues

We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalue-optimization, i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrix-valued function. We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue-optimization. In the spectrum of an optimal matrix, the kth and (k + 1)st largest eigenvalues tend to be equal and frequently have multiplicity greater than two. This clustering is intuitively plausible and has been observed as early as 1975. When the matrix-valued function is affine, we prove that clustering must occur at extreme points of the set of optimal solutions, if the number of variables is sufficiently large. We also give a lower bound on the multiplicity of the critical eigenvalue. These results generalize to the case of a general matrix-valued function under appropriate conditions.

Octane: A New Heuristic for Pure 0-1 Programs

We propose a new heuristic for pure 0--1 programs, which finds feasible integer points by enumerating extended facets of the octahedron, the outer polar of the unit hypercube. We give efficient algorithms to carry out the enumeration, and we explain how our heuristic can be embedded in a branch-and-cut framework. Finally, we present computational results on a set of pure 0--1 programs taken from MIPLIB and other sources.

The Geometry of Semidefinite Programming

Consider the primal-dual pair of optimization problems

On the Closedness of the Linear Image of a Closed Convex Cone

\begin{gathered} Min \left\langle {c,x} \right\rangle {\rm M}ax \left\langle {b,y} \right\rangle \hfill \\ (P) s.t. x \in K s.t. z \in K* (D) \hfill \\ Ax = b A*y + z = c \hfill \\ \end{gathered}

Cone-LP's and semidefinite programs: Geometry and a simplex-type method

where X and Y are Euclidean spaces with dim X ≥ dim Y. A : X → Y is a linear operator, assumed to be onto. A* : Y → X is its adjoint. K is a closed, convex, facially exposed cone in X. K* := {z|〈z,x〉≤ 0 ∀x∈K} is the dual of K, also a closed, convex, facially exposed cone.

On the generic properties of convex optimization problems in conic form

The active field of Functional Data Analysis (about understanding the variation in a set of curves) has been recently extended to Object Oriented Data Analysis, which considers populations of more general objects. A particularly challenging extension of this set of ideas is to populations of tree-structured objects. We develop an analog of Principal Component Analysis for trees, based on the notion of tree-lines, and propose numerically fast (linear time) algorithms to solve the resulting optimization problems. The solutions we obtain are used in the analysis of a data set of 73 individuals, where each data object is a tree of blood vessels in one persons brain.

Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals

When is the linear image of a closed convex cone closed? We present very simple and intuitive necessary conditions that (1) unify, and generalize seemingly disparate, classical sufficientconditions such as polyhedrality of the cone, and Slater-type conditions; (2) are necessary and sufficient, when the dual cone belongs to a class that we call nice cones (nice cones subsume all cones amenable to treatment by efficient optimization algorithms, for instance, polyhedral, semidefinite, and p-cones); and (3) provide similarly attractive conditions for an equivalent problem: the closedness of the sum of two closed convex cones.

Solving Integer and Disjunctive Programs by Lift and Project

We designed a simple computational exercise to compare weak and strong integer programming formulations of the traveling salesman problem. Using commercial IP software, and a short (60 line long) MATLAB code, students can optimally solve instances with up to 70 cities in a few minutes by adding cuts from the stronger formulation to the weaker, but simpler one.