# Abe Engle; Local Convergence Rates of a Gauss-Newton Method for Convex Compositions

Mar 14, 2017, 4pm
PDL C-401
Abe Engle, Department of Mathematics, University of Washington

Abstract: We discuss a Gauss-Newton methodology for minimizing convex compositions of smooth functions. We analyze current local rates of quadratic convergence when the subproblems are exactly solved and propose inexact methods that relax current sharpness assumptions while maintaining speeds of convergence.

# Rebecca Hoberg; Number balancing is as hard as Minkowski’s theorem

Feb 21, 2017, 4pm
PDL C-401
Rebecca Hoberg, Department of Mathematics, University of Washington

Abstract: The number balancing problem (NBP) is the following: given real numbers $a_1,…,a_n \in [0,1]$, find two distinct subsets of $[n]$ so that the difference of the sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most $O(\frac{\sqrt{n}}{2^n})$. Finding the minimum, however, is NP-hard. In polynomial time, the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most $n^{-\Theta(\log n)}$, but no further improvement has been made since then.

In this talk, we show that an approximate oracle for Minkowski’s Theorem gives an approximate NBP oracle, and conversely an approximate NBP oracle gives an approximate Minkowski oracle. In particular, we prove that any polynomial time algorithm that guarantees a NBP solution of difference at most $O(\frac{2^\sqrt{n}}{2^n})$ would give a polynomial approximation for Minkowski as well as a polynomial factor approximation algorithm for the Shortest Vector Problem. This is joint work with Harish Ramadas, Thomas Rothvoss, and Xin Yang.

# Lynn Chua; Gram spectrahedra

Feb 14, 2017, 4pm
THO 234
Lynn Chua, Department of Computer Science, UC Berkeley

Abstract: Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. This is a fundamental object in polynomial optimization and convex algebraic geometry. We summarize results on sums of squares that fit naturally into the context of Gram spectrahedra, present some new results, and highlight related open questions. We discuss sum-of-squares representations of minimal length and relate them to Hermitian Gram spectrahedra and point evaluations on toric varieties.

# Jon Lee; Mixed-Integer Nonlinear Optimization: Let’s get crazy!

CORE Series
Tuesday, Jan 31, 2017
LOW 105, 4:00-5:00PM
Jon Lee, University of Michigan

TITLE: Mixed-Integer Nonlinear Optimization: Let’s get crazy!

ABSTRACT: Mixed-Integer Nonlinear Optimization (MINLP) is the mother of all (deterministic) optimization problems. As such, it is too broad a category to do something for in general. Still, there are broad classes where we have positive complexity results, as well as narrow classes where we have negative complexity results. I will sample some of the landscape.

Considering very damning complexity results, we should have modest goals on the computational side. Nonetheless, there has been some substantial success with “general-purpose” algorithms and software for MINLP, applicable to rather broad (and practically relevant) formulations. In particular, for formulations with convex relaxations we have Bonmin (which implements NLP-based branch-and-bound and outer approximation), and for “factorable” formulations with non-convex relaxations we have Baron, Scip, Antigone, and Couenne (which implement spatial branch-and-bound). Still, the situation is not that we have extremely reliable nor scalable technology (in sharp contrast with LP and in less sharp contrast to MILP). Much of the research effort in modeling and algorithm improvement relates to finding tight convexifications. I will present recent mathematical efforts that I have been involved in to make algorithmic approaches to MILP more efficient and robust. In doing so, many areas of mathematics surprisingly pop up. Substantial parts of this are joint works with Daphne Skipper (U.S. Naval Academy) and with Emily Speakman (U. Michigan).

BIO:  Jon Lee is the G. Lawton and Louise G. Johnson Professor of Engineering at the University of Michigan. He received his Ph.D. from Cornell University. Jon has previously been a faculty member at Yale University and the University of Kentucky, and an adjunct professor at New York University.  He was a Research Staff member at the IBM T.J. Watson Research Center, where he managed the mathematical programming group. Jon is author of ~120 papers, the text “A First Course in Combinatorial Optimization” (Cambridge University Press), and the open-source book “A First Course in Linear Optimization” (Reex Press). He was the founding Managing Editor of the journal Discrete Optimization (2004-06), he is currently co-Editor of the journal Mathematical Programming, and he is on the editorial boards of the journals Optimization and Engineering, and Discrete Applied Mathematics. Jon was Chair of the Executive Committee of the Mathematical Optimization Society (2008-10), and Chair of the INFORMS Optimization Society (2010-12). He was awarded the INFORMS Computing Society Prize (2010), and he is a Fellow of INFORMS (since 2013).

# Amy Wiebe; On representing the positive semidefinite cone using the second order cone

Jan 17, 2017, 4pm
PDL C-401
Amy Wiebe, Department of Mathematics, University of Washington

Abstract:  In this talk we present a recent result of Hamza Fawzi which answers the question of whether it is possible to express the general positive semidefinite cone using second-order cones. The result shows that the 3 x 3 positive semidefinite cone S^3_+ does not admit a second-order cone representation. The proof relies on the method of Gouveia, Parrilo and Thomas which shows that existence of cone lifts of convex sets is equivalent to a certain factorization of the slack operator of the set. We explain how this framework is used in the paper to show that S^3_+ has no finite second order cone lift.

# Winter 2017 Calendar

Jan 17
Amy Wiebe, Department of Mathematics, University of Washington
On representing the positive semidefinite cone using the second order cone

Jan 31 [CORE]
Jon Lee, Department of Industrial and Operations Engineering, University of Michigan
Mixed-Integer Nonlinear Optimization: Let’s get crazy!

Feb 7
Asen L. DontchevMathematical Reviews and University of Michigan
Strong Metric Subregularity

Feb 14
Lynn Chua, Department of Computer Science, UC Berkeley
Gram spectrahedra

Feb 21
Rebecca Hoberg, Department of Mathematics, University of Washington
Number balancing is as hard as Minkowski’s theorem

Mar 14
Abe Engle, Department of Mathematics, University of Washington
Local Convergence Rates of a Gauss-Newton Method for Convex Compositions

# Asen Dontchev; Strong Metric Subregularity

Feb 7, 2017, 4pm
THO 234
Asen Dontchev, Mathematical Reviews and University of Michigan

Abstract:  Although the property of strong metric sub regularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong metric regularity. In this talk I will try to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. In particular, it obeys the inverse function theorem paradigm also for nonsmooth perturbations.

A sufficient condition  for strong metric subregularity is established in terms of surjectivity of the Fr\’echet coderivative will be presented, and it will be shown by a counterexample that subjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton’s method for solving variational inequalities will be considered including  inexact and semismooth methods. A characterization of the strong metric subregularity of the KKT mapping will be demonstrated, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.