Control Theory and Optimization
IL16.1
Recent results around the diameter of polyhedra Friedrich Eisenbrand
EPFL, Switzerland friedrich.eisenbrand@epfl.ch
2010Mathematics Subject Classification. 52B11, 52B55
Keywords. Convex Geometry, Linear Programming, Polyhedra, Algorithms
The diameter of a polyhedronPis the largest distance of a pair of vertices in the edge-graph ofP. The question whether the diameter of a polyhedron can be bounded by a polynomial in the dimension and number of facets ofP remains one of most important open problems in convex geometry.
In the last three years, there was an accelerated interest in this famous open problem which has lead to many interesting results and techniques, also due to a celebrated breakthrough of Santos disproving the Hirsch conjecture. Here, I want to describe a subset of these recent results and describe some open problems.
IL16.5
Optimization over polynomials: selected topics Monique Laurent
Centrum Wiskunde & Informatica and Tilburg University, Netherlands [email protected]
2010Mathematics Subject Classification. 90C22, 90C27, 90C30, 44A60, 13J30
Keywords. Positive polynomial, sum of squares, moment problem, combinatorial optimization, semi- definite optimization
Minimizing a polynomial function over a region defined by polynomial inequalities models broad classes of hard problems from combinatorics, geometry and optimization. New algo- rithmic approaches have emerged recently for computing the global minimum, by combining tools from real algebra (sums of squares of polynomials) and functional analysis (moments of measures) with semidefinite optimization. Sums of squares are used to certify positive poly- nomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this gen- eral methodology, illustrate how it applies to some combinatorial graph problems, and discuss links with other relaxation methods.
IL16.3
Nonsmooth optimization: conditioning, convergence, and semi-algebraic models
Adrian Lewis
Cornell University, United States of America [email protected]
www.icm2014.org 119
International Congress of Mathematicians, Seoul, 2014
2010Mathematics Subject Classification. 90C31, 49K40, 65K10, 14P10, 93D20
Keywords. Variational analysis, nonsmooth optimization, inverse function, alternating projections, met- ric regularity, semi-algebraic, convergence rate, condition number, normal cone, transversality, quasi- Newton, eigenvalue optimization, identifiable manifold
Variational analysis has come of age. Long an elegant theoretical toolkit for variational math- ematics and nonsmooth optimization, it now increasingly underpins the study of algorithms, and a rich interplay with semi-algebraic geometry illuminates its generic applicability. As an example, alternating projections - a rudimentary but enduring algorithm for exploring the intersection of two arbitrary closed sets - concisely illustrates several far-reaching and inter- dependent variational ideas. A transversality measure, intuitively an angle and generically nonzero, controls several key properties: the method’s linear convergence rate, a posteriori error bounds, sensitivity to data perturbations, and robustness relative to problem description.
These linked ideas emerge in a wide variety of computational problems. Optimization in par- ticular is rich in examples that depend, around critical points, on “active” manifolds of nearby approximately critical points. Such manifolds, central to classical theoretical and computa- tional optimization, exist generically in the semi-algebraic case. We discuss examples from eigenvalue optimization and stable polynomials in control systems, and a prox-linear algo- rithm for large-scale composite optimization applications such as machine learning.
IL16.4
Carleman estimates, results on control and stabilization for partial differential equations
Luc Robbiano
Université de Versailles Saint-Quentin-en-Yvelines, France [email protected]
2010Mathematics Subject Classification. 35A02, 35Q93, 93B05, 93D15
Keywords. Carleman estimates, control, null control, stabilization, exterior problem
In this survey we give some results based on Carleman estimates. We recall the classical uniqueness result based on interior Carleman estimate. We give Carleman estimate up the boundary useful for the applications. The main applications are, approximative control for wave equation, null control for heat equation, stabilization for wave equation for an interior damping or for a boundary damping and local energy decay for wave equation in exterior domain.
IL16.2
Models and feedback stabilization of open quantum systems Pierre Rouchon
Mines Paris Tech, France [email protected]
2010Mathematics Subject Classification. 93B52, 93D15, 81V10, 81P15
120 www.icm2014.org
Control Theory and Optimization
Keywords. Markov model, open quantum system, quantum filtering, quantum feedback, quantum mas- ter equations
At the quantum level, feedback-loops have to take into account measurement back-action. We present here the structure of the Markovian models including such back-action and sketch two stabilization methods: measurement-based feedback where an open quantum system is stabilized by a classical controller; coherent or autonomous feedback where a quantum system is stabilized by a quantum controller with decoherence (reservoir engineering). We begin to explain these models and methods for the photon box experiments realized in the group of Serge Haroche (Nobel Prize 2012). We present then these models and methods for general open quantum systems.
IL16.6
Time-inconsistent optimal control problems Jiongmin Yong
University of Central Florida, United States of America [email protected]
2010Mathematics Subject Classification. 93E20, 49L20, 49N05, 49N70
Keywords. Stochastic optimal control, time inconsistency, equilibrium solution, Hamilton-Jacobi-Bell- man equation, differential games
An optimal control problem is time-consistent if for any initial pair of time and state, whenever there exists an optimal control, it will stay optimal thereafter. In real world, however, such kind of time-consistency is hardly true, mainly due to the time-inconsistency of decision maker’s time-preference and/or risk-preference. In another word, most optimal control problems, if not all, are not time-consistent, or time-inconsistent. In this paper, some general time-inconsistent optimal control problems are formulated for stochastic differential equations. Recent works of the author concerning the (time-consistent) equilibrium solutions to the time-inconsistent problems are surveyed.
www.icm2014.org 121