Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worstcase objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.
The bounded degree sum-of-squares (BSOS) hierarchy of Lasserre et al. (EURO J Comput Optim 1-31, 2015) constructs lower bounds for a general polynomial optimization problem with compact feasible set, by solving a sequence of semi-definite programming (SDP) problems. Lasserre, Toh, and Yang prove that these lower bounds converge to the optimal value of the original problem, under some assumptions. In this paper, we analyze the BSOS hierarchy and study its numerical performance on a specific class of bilinear programming problems, called pooling problems, that arise in the refinery and chemical process industries.
Uncertain partially observable Markov decision processes (uPOMDPs) allow the probabilistic transition and observation functions of standard POMDPs to belong to a so-called uncertainty set.
Such uncertainty, referred to as epistemic uncertainty, captures uncountable sets of probability distributions caused by, for instance, a lack of data available.
We develop an algorithm to compute finite-memory policies for uPOMDPs that robustly satisfy specifications against any admissible distribution.
In general, computing such policies is theoretically and practically intractable.
We provide an efficient solution to this problem in four steps.
(1) We state the underlying problem as a nonconvex optimization problem with infinitely many constraints.
(2) A dedicated dualization scheme yields a dual problem that is still nonconvex but has finitely many constraints.
(3) We linearize this dual problem and (4) solve the resulting finite linear program to obtain locally optimal solutions to the original problem.
The resulting problem formulation is exponentially smaller than those resulting from existing methods.
We demonstrate the applicability of our algorithm using large instances of an aircraft collision-avoidance scenario and a novel spacecraft motion planning case study.
The sparse bounded degree sum-of-squares (sparse-BSOS) hierarchy of Weisser, Lasserre and Toh [arXiv:1607.01151,2016] constructs a sequence of lower bounds for a sparse polynomial optimization problem. Under some assumptions, it is proven by the authors that the sequence converges to the optimal value. In this paper, we modify the hierarchy to deal with problems containing equality constraints directly, without eliminating or replacing them by two inequalities. We also evaluate the sparse-BSOS hierarchy on a well-known bilinear programming problem, called the pooling problem.
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