Relax, no need to round: integrality of clustering formulations

We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: $k$-means and $k$-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are $k$ clusters in $\mathbb{R}^m$ and data from each cluster consists of $n$ points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these $k$ clusters as the optimal integral solution? For the $k$-median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation $\epsilon > 0$ between the balls. In other words, the pairwise center separation is $\Delta > 2+\epsilon$. Under the same distributional model, the $k$-means LP relaxation fails to recover such clusters at separation as large as $\Delta = 4$. Yet, if we enforce PSD constraints on the $k$-means LP, we get exact cluster recovery at center separation $\Delta > 2\sqrt2(1+\sqrt{1/m})$. In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the $k$-means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.Comment: 30 pages, ITCS 201

Hierarchies of Relaxations for Online Prediction Problems with Evolving Constraints

We study online prediction where regret of the algorithm is measured against a benchmark defined via evolving constraints. This framework captures online prediction on graphs, as well as other prediction problems with combinatorial structure. A key aspect here is that finding the optimal benchmark predictor (even in hindsight, given all the data) might be computationally hard due to the combinatorial nature of the constraints. Despite this, we provide polynomial-time \emph{prediction} algorithms that achieve low regret against combinatorial benchmark sets. We do so by building improper learning algorithms based on two ideas that work together. The first is to alleviate part of the computational burden through random playout, and the second is to employ Lasserre semidefinite hierarchies to approximate the resulting integer program. Interestingly, for our prediction algorithms, we only need to compute the values of the semidefinite programs and not the rounded solutions. However, the integrality gap for Lasserre hierarchy \emph{does} enter the generic regret bound in terms of Rademacher complexity of the benchmark set. This establishes a trade-off between the computation time and the regret bound of the algorithm

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York

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 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

Integer Programming Approaches for Some Non-convex and Stochastic Optimization Problems

In this dissertation we study several non-convex and stochastic optimization problems. The common theme is the use of mixed-integer programming (MIP) techniques including valid inequalities and reformulation to solve these problems. We first study a strategic capacity planning model which captures the trade-off between the incentive to delay capacity installation to wait for improved technology and the need for some capacity to be installed to meet current demands. This problem is naturally formulated as a MIP with a bilinear objective. We develop several linear MIP formulations, along with classes of strong valid inequalities. We also present a specialized branch-and-cut algorithm to solve a compact concave formulation. Computational results indicate that these formulations can be used to solve large-scale instances. We next study methods for optimization with joint probabilistic constraints. These problems are challenging because evaluating solution feasibility requires multidimensional integration and the feasible region is not convex. We propose and analyze a Monte Carlo sampling scheme to simplify the probabilistic structure of such problems. Computational tests of the approach indicate that it can yield good feasible solutions and reasonable bounds on their quality. Next, we study a MIP formulation of the non-convex sample approximation problem. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. Computational results indicate that large-scale instances can be solved using the strengthened formulations. Finally, we study optimization problems with stochastic dominance constraints. A stochastic dominance constraint states that a random outcome which depends on the decision variables should stochastically dominate a given random variable. We present new formulations for both first and second order stochastic dominance which are significantly more compact than existing formulations. Computational tests illustrate the benefits of the new formulations.Ph.D.Committee Co-Chair: Ahmed, Shabbir; Committee Co-Chair: Nemhauser, George; Committee Member: Cook, Bill; Committee Member: Gu, Zonghao; Committee Member: Parker, R. Gar