Polynomial Approximations for Infinite-Dimensional Optimization Problems

Many real-life decision problems in management science and engineering involve decisions that are functions of time and/or uncertainty. The resulting optimization models are therefore naturally formulated on infinite-dimensional function spaces. However, such infinite-dimensional optimization problems are notoriously difficult, and to solve them one usually has to resort to approximation methods. The objective of this thesis is to devise polynomial approximations for solving continuous linear programs and multi-stage stochastic programs, both of which constitute important classes of infinite-dimensional optimization problems with manifold practical applications. Approximating the functional decision variables by polynomials allows us to apply sum-of-squares techniques from algebraic geometry to reformulate the resulting problems as tractable semidefinite programs, which can be solved efficiently with interior point algorithms. Continuous linear programs represent deterministic optimization problems whose decision variables are functions of time subject to pointwise and dynamic linear constraints. They have attracted considerable interest due to their potential for modelling manufacturing, scheduling and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (non-separated) problem instances. In this thesis we propose a more generic approximation scheme for non-separated continuous linear programs, which are believed to be NP-hard. We approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. To estimate the approximation error, we also compute a lower bound by solving a dual continuous linear program in (piecewise) polynomial decision rules. Multi-stage stochastic programming provides a versatile framework for optimal decision making under uncertainty, but it gives rise to hard functional optimization problems since the adaptive recourse decisions must be modelled as functions of some or all uncertain parameters. We propose to approximate these recourse decisions by polynomial decision rules and show that the best polynomial decision rule of a fixed degree can be computed efficiently. Again, we show that the suboptimality of the best polynomial decision rule can be estimated efficiently by solving a dual version of the stochastic program in polynomial decision rules. Recent progress in the theory of dynamic risk measures has found a strong echo in stochastic programming, where the time-consistency of dynamic decision making under uncertainty is currently under scrutiny. We extend the concepts of coherence and time consistency to stochastic programming models subject to distributional ambiguity, which motivates us to introduce robust dynamic risk measures. We discuss conditions under which these robust risk measures inherit coherence and time-consistency from their nominal counterparts. We also propose an approximation scheme based on polynomial decision rules for solving linear multi-stage stochastic programs involving robust dynamic risk measures

A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets \K (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as pseudorandom generators for halfspaces. In this work, we investigate the limits of these techniques by proving inapproximability results for the sign function. Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be learned with respect to log-concave distributions on $\mathbb{R}^n$ in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. We ask whether this technique can be extended beyond log-concave distributions, and establish a negative result. We show that polynomials of any degree cannot approximate the sign function to within arbitrarily low error for a large class of non-log-concave distributions on the real line, including those with densities proportional to $\exp(-|x|^{0.99})$. Secondly, we investigate the derandomization of Chernoff-type concentration inequalities. Chernoff-type tail bounds on sums of independent random variables have pervasive applications in theoretical computer science. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that these inequalities can be established for sums of random variables with only $O(\log(1/\delta))$-wise independence, for a tail probability of $\delta$. We show that their results are tight up to constant factors. These results rely on techniques from weighted approximation theory, which studies how well functions on the real line can be approximated by polynomials under various distributions. We believe that these techniques will have further applications in other areas of computer science.Comment: 22 page