72 research outputs found
Regret Models and Preprocessing Techniques for Combinatorial Optimization under Uncertainty
Ph.DDOCTOR OF PHILOSOPH
A Unified Theory of Robust and Distributionally Robust Optimization via the Primal-Worst-Equals-Dual-Best Principle
Robust and distributionally robust optimization are modeling paradigms for
decision-making under uncertainty where the uncertain parameters are only known
to reside in an uncertainty set or are governed by any probability distribution
from within an ambiguity set, respectively, and a decision is sought that
minimizes a cost function under the most adverse outcome of the uncertainty. In
this paper, we develop a rigorous and general theory of robust and
distributionally robust nonlinear optimization using the language of convex
analysis. Our framework is based on a generalized
`primal-worst-equals-dual-best' principle that establishes strong duality
between a semi-infinite primal worst and a non-convex dual best formulation,
both of which admit finite convex reformulations. This principle offers an
alternative formulation for robust optimization problems that obviates the need
to mobilize the machinery of abstract semi-infinite duality theory to prove
strong duality in distributionally robust optimization. We illustrate the
modeling power of our approach through convex reformulations for
distributionally robust optimization problems whose ambiguity sets are defined
through general optimal transport distances, which generalize earlier results
for Wasserstein ambiguity sets.Comment: Previous title: Mathematical Foundations of Robust and
Distributionally Robust Optimizatio
Prepare for the Expected Worst: Algorithms for Reconfigurable Resources Under Uncertainty
In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study variants of minimum spanning and Steiner trees, minimum cuts, and facility location. Up to constants, our approximation guarantees match those of previously-studied algorithms for demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems
Distributionally robust L1-estimation in multiple linear regression
Linear regression is one of the most important and widely used techniques in data analysis, for which a key step is the estimation of the unknown parameters. However, it is often carried out under the assumption that the full information of the error distribution is available. This is clearly unrealistic in practice. In this paper, we propose a distributionally robust formulation of L1-estimation (or the least absolute value estimation) problem, where the only knowledge on the error distribution is that it belongs to a well-defined ambiguity set. We then reformulate the estimation problem as a computationally tractable conic optimization problem by using duality theory. Finally, a numerical example is solved as a conic optimization problem to demonstrate the effectiveness of the proposed approach
Ergodic Control and Polyhedral approaches to PageRank Optimization
We study a general class of PageRank optimization problems which consist in
finding an optimal outlink strategy for a web site subject to design
constraints. We consider both a continuous problem, in which one can choose the
intensity of a link, and a discrete one, in which in each page, there are
obligatory links, facultative links and forbidden links. We show that the
continuous problem, as well as its discrete variant when there are no
constraints coupling different pages, can both be modeled by constrained Markov
decision processes with ergodic reward, in which the webmaster determines the
transition probabilities of websurfers. Although the number of actions turns
out to be exponential, we show that an associated polytope of transition
measures has a concise representation, from which we deduce that the continuous
problem is solvable in polynomial time, and that the same is true for the
discrete problem when there are no coupling constraints. We also provide
efficient algorithms, adapted to very large networks. Then, we investigate the
qualitative features of optimal outlink strategies, and identify in particular
assumptions under which there exists a "master" page to which all controlled
pages should point. We report numerical results on fragments of the real web
graph.Comment: 39 page
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