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68 OP14 Abstracts 68 OP14 Abstracts IP1 of polyhedra. Universal Gradient Methods Francisco Santos In Convex Optimization, numerical schemes are always University of Cantabria developed for some specific problem classes. One of the [email protected] most important characteristics of such classes is the level of smoothness of the objective function. Methods for nons- IP4 mooth functions are different from the methods for smooth ones. However, very often the level of smoothness of the Combinatorial Optimization for National Security objective is difficult to estimate in advance. In this talk Applications we present algorithms which adjust their behavior in ac- cordance to the actual level of smoothness observed during National-security optimization problems emphasize safety, the minimization process. Their only input parameter is cost, and compliance, frequently with some twist. They the required accuracy of the solution. We discuss also the may merit parallelization, have to run on small, weak plat- abilities of these schemes in reconstructing the dual solu- forms, or have unusual constraints or objectives. We de- tions. scribe several recent such applications. We summarize a massively-parallel branch-and-bound implementation for a Yurii Nesterov core task in machine classification. A challenging spam- CORE classification problem scales perfectly to over 6000 pro- Universite catholique de Louvain cessors. We discuss math programming for benchmark- [email protected] ing practical wireless-sensor-management heuristics. We sketch theoretically justified algorithms for a scheduling problem motivated by nuclear weapons inspections. Time IP2 permitting, we will present open problems. For exam- ple, can modern nonlinear solvers benchmark a simple lin- Optimizing and Coordinating Healthcare Networks earization heuristic for placing imperfect sensors in a mu- and Markets nicipal water network? Healthcare systems pose a range of major access, quality Cynthia Phillips and cost challenges in the U.S. and globally. We survey Sandia National Laboratories several healthcare network optimization problems that are [email protected] typically challenging in practice and theory. The challenge comes from network affects, including the fact that in many cases the network consists of selfish and competing players. IP5 Incorporating the complex dynamics into the optimization The Euclidean Distance Degree of an Algebraic Set is rather essential, but hard to models and often leads to computationally intractable models. We focus attention to It is a common problem in optimization to minimize the computationally tractable and practical policies, and an- Euclidean distance from a given data point u to some set alyze their worst-case performance compared to optimal X. In this talk I will consider the situation in which X is de- policies that are computationally intractable and concep- fined by a finite set of polynomial equations. The number tually impractical. of critical points of the objective function on X is called the Euclidean distance degree of X, and is an intrinsic measure Retsef Levi of the complexity of this polynomial optimization prob- Massachusetts Institute of Technology lem. Algebraic geometry offers powerful tools to calculate [email protected] this degree in many situations. I will explain the algebraic methods involved, and illustrate the formulas that can be obtained in several situations ranging from matrix analy- IP3 sis to control theory to computer vision. Joint work with Recent Progress on the Diameter of Polyhedra and Jan Draisma, Emil Horobet, Giorgio Ottaviani and Bernd Simplicial Complexes Sturmfels. The Hirsch conjecture, posed in 1957, stated that the graph Rekha Thomas of a d-dimensional polytope or polyhedron with n facets University of Washington cannot have diameter greater than n − d. The conjec- [email protected] ture itself has been disproved (Klee-Walkup (1967) for un- bounded polyhedra, Santos (2010) for bounded polytopes), but what we know about the underlying question is quite IP6 scarce. Most notably, no polynomial upper bound is known Modeling Wholesale Electricity Markets with Hy- for the diameters that were conjectured to be linear. In dro Storage contrast, no polyhedron violating the Hirsch bound by more than 25% is known. In this talk we review several re- Over the past two decades most industrialized nations have cent attempts and progress on the question. Some of these instituted markets for wholesale electricity supply. Opti- work in the world of polyhedra or (more often) bounded mization models play a key role in these markets. The un- polytopes, but some try to shed light on the question by derstanding of electricity markets has grown substantially generalizing it to simplicial complexes. In particular, we through various market crises, and there is now a set of show that the maximum diameter of arbitrary simplicial standard electricity market design principles for systems complexesisinnΘ(d), we sketch the proof of Hirsch’s bound with mainly thermal plant. On the other hand, markets for “flag’ polyhedra (and more general objects) by Adipr- with lots of hydro storage can face shortage risks, leading to asito and Benedetti, and we summarize the main ideas in production and pricing arrangements in these markets that the polymath 3 project, a web-based collective effort try- vary widely across jurisdictions. We show how stochastic ing to prove an upper bound of type nd for the diameters optimization and complementarity models can be used to OP14 Abstracts 69 improve our understanding of these systems. method is strongly polynomial for solving deterministic MDPs regardless of discount factors. Andy Philpott University of Auckland Yinyu Ye [email protected] Stanford University [email protected] IP7 Large-Scale Optimization of Multidisciplinary En- CP1 gineering Systems Superlinearly Convergent Smoothing Continuation Algorithms for Nonlinear Complementarity Prob- There is a compelling incentive for using optimization to lems over Definable Convex Cones design aerospace systems due to large penalties incurred by their weight. The design of these systems is challeng- We consider the superlinear convergence of smoothing con- ing due to their complexity. We tackle these challenges by tinuation algorithms without Jacobian consistency. In our developing a new view of multidisciplinary systems, and approach, we use a barrier based smoothing approxima- by combining gradient-based optimization, efficient gradi- tion, which is defined for every closed convex cone. When ent computation, and Newton-type methods. Our applica- the barrier has definable derivative, we prove the superlin- tions include wing design based on Navier–Stokes aerody- ear convergence of a smoothing continuation algorithm for namic models coupled to finite-element structural models, solving nonlinear complementarity problems over the cone. and satellite design including trajectory optimization. The Such barriers exist for cones definable in the o-minimal ex- methods used in this work are generalized and proposed as pansion of globally analytic sets by power functions with a new framework for solving large-scale optimization prob- real algebraic exponents. lems. Chek Beng Chua Joaquim Martins Nanyang Technological University University of Michigan [email protected] [email protected] CP1 IP8 On the Quadratic Eigenvalue Complementarity A Projection Hierarchy for Some NP-hard Opti- Problem mization Problems A new sufficient condition for the existence of solutions to There exist several hierarchies of relaxations which allow the Quadratic Eigenvalue Complementarity Problem (QE- to solve NP-hard combinatorial optimization problems to iCP) is established. This condition exploits the reduction optimality. The Lasserre and Parrilo hierarchies are based of QEiCP to a normal EiCP. An upper bound for the num- on semidefinite optimization and in each new level both the ber of solutions of the QEiCP is presented. A new strategy dimension and the number of constraints increases. As a for QEiCP is analyzed which solves the resulting EiCP by consequence even the first step up in the hierarchy leads to an equivalent Variational Inequality Problem. Numerical problems which are extremely hard to solve computation- experiments with a projection VI method illustrate the in- ally for nontrivial problem sizes. In contrast, we present terest of this methodology in practice. a hierarchy where the dimension stays fixed and only the number of constraints grows exponentially. It applies to Joaquim J. Jdice problems where the projection of the feasible set to sub- Instituto de Telecomunica˜oes problems has a ’simple’ structure. We consider this new hi- [email protected] erarchy for Max-Cut, Stable-Set and Graph-Coloring. We look at some theoretical properties, discuss practical is- Carmo Br´as sues and provide computational results, comparing the new Universidade Nova de Lisboa bounds with the current state-of-the-art. Portugal Franz Rendl [email protected] Alpen-Adria Universitaet Klagenfurt Institut fuer Mathematik Alfredo N. Iusem [email protected] IMPA, Rio de Janeiro [email protected] SP1 SIAG/OPT Prize Lecture: Efficiency of the Sim- CP1 plex and Policy Iteration Methods for Markov De- Complementarity Formulations of 0-Norm Opti- cision Processes mization Problems We prove that the classic policy-iteration method (Howard There has been interest recently in obtaining sparse solu- 1960), including the simple policy-iteration (or simplex tions to optimization problems, eg in compressed
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