# 2015 Informs Annual Meeting

SC16

INFORMS Philadelphia – 2015

4 - Quasi-newton Methods for Nonsmooth Optimization Nitish Shirish Keskar, Northwestern University, 2145 Sheridan Road, Room C210, Evanston, IL, 60208, United States of America, keskar.nitish@gmail.com, Jorge Nocedal, Andreas Waechter We study the behavior of quasi-Newton methods on large-scale nonsmooth optimization problems. We concentrate specifically on a modified BFGS approach and its limited memory implementation. We contrast this approach with other popular methods including Limited Memory Bundle Methods and Gradient Sampling methods and describe detailed numerical results from both academic and practical test problems. Finally, we briefly provide some new theoretical guarantees for this class of methods. SC16 16-Franklin 6, Marriott Efficient Algorithms for Large-Scale Convex Games Sponsor: Optimization/Linear and Conic Optimization Sponsored Session Chair: Fatma Kilinc Karzan, Assistant Professor Of Operations Research, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, United States of America, fkilinc@andrew.cmu.edu Co-Chair: Christian Kroer, PhD Student, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15213, United States of America, ckroer@cs.cmu.edu 1 - Games People (could not) Play Every two-player zero-sum game has an optimal strategy that can be found by solving an LP. But this approach is not polynomial time when the number of pure strategies for each player is exponential in the input, e.g. each player plays spanning trees of a graph. We present fast algorithms to compute Nash-equilibria for these games for bilinear payoff functions using ideas from convex and combinatorial optimization and machine learning. 2 - Maximal Cooperation in Repeated Games on Social Networks Catherine Moon, Duke University, 213 Social Sciences, Box 90097, Durham, NC, 27705, United States of America, csm17@duke.edu, Vincent Conitzer Standard results on repeated games assume that defections are instantly observable. In reality, it may take some time for agents in a network to learn that a defection has occurred. How does this affect the structure of equilibria and algorithms for computing them? In games of cooperation and defection, we prove existence of a unique maximal set of forever-cooperating agents in equilibrium, give an efficient algorithm for computing it and a condition for when the equilibrium found is credible. 3 - Faster First-order Methods for Extensive-form Game Solving Christian Kroer, PhD Student, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15213, United States of America, ckroer@cs.cmu.edu, Kevin Waugh, Fatma Kilinc Karzan, Tuomas Sandholm We investigate first-order methods for Nash equilibrium computation in zero-sum sequential games. We introduce the dilated entropy function as a distance- generating function, and develop strong-convexity bounds on this function over convex polytopes that generalize sequential-game strategy spaces. In terms of game solving, this improves the convergence rate of several first-order methods. We instantiate these methods with our results, and show that our algorithms outperform prior algorithms. 4 - Simultaneous Abstraction and Equilibrium Finding in Games Noam Brown, Carnegie Mellon University, 5000 Forbes Ave, The leading approach to solving infeasibly large imperfect-information extensive- form games is to solve a smaller abstract version of the game and then map the solution back to the original. However, it is difficult to know which actions should be included in the abstraction without first solving the game. We introduce a method that combines abstraction with equilibrium finding by enabling actions to be added to the abstraction at run time in points that the current abstraction deems important. Swati Gupta, MIT, 77 Massachusetts Avenue, E40-149, Cambridge, MA, 02139, United States of America, swatig@mit.edu Pittsburgh, PA, 15213, United States of America, noamb@andrew.cmu.edu, Tuomas Sandholm

We present a partition-and-bound method for multistage adaptive mixed integer optimization problems that extends previous work on finite adaptability. The method analyzes the solution to a non-adaptive problem to determine how the uncertainty set is restricting the objective, and uses this to partition the set. A lower bound is calculated, and the partitioning is repeated until a desired gap is reached. We demonstrate in computational experiments that the method produces good solutions quickly. 2 - The Role of Robustness in Modern Statistical Regression Martin Copenhaver, MIT, 77 Massachusetts Avenue, E40-135, Cambridge, MA, 02139, United States of America, mcopen@mit.edu, Dimitris Bertsimas Sparsity is a key driver in modern statistical estimation problems, yet reliably sparse solutions remain elusive. Despite this, many regularization methods often perform well in the face of noise in the data. In the domains of linear, median, and matrix regression, we characterize precisely when regularization problems correspond to simple robust optimization problems. In doing so, we contend that it is robustness, not sparsity, which is critical to the success of modern statistical methods. 3 - Robust Ambulance Deployment Yee Sian Ng, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA, 02139, United States of America, yeesian@mit.edu, Dimitris Bertsimas In Emergency Medical Systems (EMS), operators deploy a fleet of ambulances to a set of locations before dispatching them in response to emergency calls. The goal is to minimize the fraction of calls with late response times, which we formulate as a two-stage Robust Optimization model with integer recourse. We propose a practical algorithm for solving the model, and demonstrate the benefits of this approach in a realistic case study of the EMS in Washington DC. SC15 15-Franklin 5, Marriott Nonlinear Optimization Methods with Stochastic and Second Order Information Sponsor: Optimization/Nonlinear Programming Sponsored Session Chair: Katya Scheinberg, Lehigh University, 200 W Packer ave, Bethlehem, PA, 18015, United States of America, katyas@lehigh.edu 1 - Stochastic Optimization using a Trust-region Method and Random Models Matt Menickelly, Lehigh University, 200 W. Packer Ave., Bethlehem, PA, 18015, United States of America, mjm412@lehigh.edu, Katya Scheinberg, Ruobing Chen We propose and analyze a trust-region model-based method for solving unconstrained stochastic optimization problems. The convergence analysis relies only on requirements that random models and point estimates are sufficiently accurate with sufficiently high, but fixed, probability. We present examples of generating sufficiently accurate random models under different noise assumptions and compare with existing approaches based on sample averaging or stochastic gradients. 2 - Accelerated Proximal Quasi-newton Algorithm with Global Complexity Analysis Hiva Ghanbari, Lehigh University, Harold S.Mohler Laboratory,, 200 W. Packer Ave., Bethlehem, PA, 18015, United States of America, hig213@lehigh.edu, Katya Scheinberg We propose a proximal quasi-newton algorithm in the spirit of the Nesterov’s accelerated scheme for solving convex composite optimization problems. We present the analysis reflecting the convergence properties of the proposed algorithm, which preserves the sublinear rate of convergence under some mild assumptions. Furthermore, we show that, under careful use of second order information, this method obtains the same rate of convergence of Accelerated Proximal Gradient Algorithm. 3 - Selective Linearization Algorithm for Multi-block Convex Optimization Yu Du, Rutgers University, 100 Rockafeller Road, Rutgers business school, Piscataway, NJ, 08854, United States of America, duyu197@gmail.com, Andrzej Ruszczynski, Xiaodong Lin We consider the problem of minimizing the multi-block structured convex non- smooth functions. We introduce the Selective Linearization(SLIN) Algorithm which iteratively solves a series of subproblems by linearizing some blocks and approaches the optimal solution. Global convergence is achieved and SLIN algorithm is proven to converge at rate of O(1/k). We apply the SLIN algorithm on structured regularization applications, showing fast running time for large scale problems.

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