Exploiting problem structure in optimization under uncertainty via online convex optimization

Abstract

In this paper, we consider two paradigms that are developed to account for uncertainty in optimization models: robust optimization (RO) and joint estimation-optimization (JEO). We examine recent developments on efficient and scalable iterative first-order methods for these problems, and show that these iterative methods can be viewed through the lens of online convex optimization (OCO). The standard OCO framework has seen much success for its ability to handle decision-making in dynamic, uncertain, and even adversarial environments. Nevertheless, our applications of interest present further flexibility in OCO via three simple modifications to standard OCO assumptions: we introduce two new concepts of weighted regret and online saddle point problems and study the possibility of making lookahead (anticipatory) decisions. Our analyses demonstrate that these flexibilities introduced into the OCO framework have significant consequences whenever they are applicable. For example, in the strongly convex case, minimizing unweighted regret has a proven optimal bound of O(log(T )/T ), whereas we show that a bound of O(1/T ) is possible when we consider weighted regret. Similarly, for the smooth case, considering 1-lookahead decisions results in a O(1/T ) bound, compared to O(1/ √ T ) in the standard OCO setting. Consequently, these OCO tools are instrumental in exploiting structural properties of functions and results in improved convergence rates for RO and JEO. In certain cases, our results for RO and JEO match the best known or optimal rates in the corresponding problem classes without data uncertainty.

Robust Optimization for the Pooling Problem

“…Furthermore, even finding a feasible solution to Problem (1) requires solving this non-convex problem to global optimality. Problems in these categories can generally only be solved using semiinfinite programming techniques [46][47][48] , which are limited to small scale problems, or approximate robust optimization schemes [49][50][51][52] . Category 2 has received limited attention in the robust optimization and semiinfinite programming communities [53][54][55] .…”

Duality-driven optimization in energy management: offline and online algorithms for resource allocation problems

“…In any case, the classical OCO framework does not consider the availability of further information (e.g., obtained from measurements) on the part of the objective function corresponding to the current stage. It is only very recently that OCO-models are studied with so-called 1-step look-ahead features, where such measurements are (partly) available (see, e.g., [73]).…”

Scaling Up Dynamic Optimization Problems: A Divide-and-Conquer Approach

“…To do so, the tracker with the second best found position's fitness value f (g ⋆ ) will be removed. For determining tracker populations which are under exclusion condition, the Euclidean distance between all pairs of trackers' g ⋆ position is calculated and compared with r excl based on (13).…”

“…Knowledge about the regularities and the structure of a problem allows us to devise more effective ways of solving them. This is common practice in many branches of optimization [10]- [13]. More recently, the term gray-box optimization has come to refer to the practice of incorporating problem structure into the optimization process [14].…”