This paper reduces the inner worst-case expectation problem exactly to a scalar budget allocation task, and embeds this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem.
Abstract
Distributionally robust optimization (DRO) with optimal transport ambiguity sets is traditionally solved by reformulating the minimax problem into a single-level convex program. While theoretically tractable, these reformulations introduce numerous auxiliary variables and demanding conic constraints that scale poorly in practice. In this paper, we address this challenge by reducing the inner worst-case expectation problem exactly to a scalar budget allocation task. This structural insight yields an efficient algorithm that bypasses large lifted reformulations, alongside a fast post-processing scheme to recover an optimal worst-case distribution supported on at most $N+1$ points, where $N$ denotes the sample size. We embed this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem. Furthermore, we extend our analysis to the dual DRO formulation, proving the existence of a least-favorable distribution supported on at most $\min\{N+n+1, KN\}$ atoms, where $n$ and $K$ denote the decision dimension and number of loss components, respectively, and provide an efficient convex programming reduction to extract it from the solution of the primal DRO. Numerical experiments demonstrate that the proposed approach significantly outperforms state-of-the-art reformulation-based solvers.
It is shown that BiCS is applicable to standard DRO, almost-sure DRO, DRO with various chance constraints, and DRO with ambiguity sets strengthened by local information, and demonstrates superior performance, including solving cases where the examined compact reformulations are unavailable or computationally difficult.
A shrinkage path heuristic is proposed that reduces the solution of a DRO problem to a one-dimensional search over the line segment connecting the sample average approximation (SAA) and the (more demanding but practically solvable) classical robust optimization solution.
Ling-Jun Meng, Ryan Cory-Wright, W. Wiesemann· 0 citations
We consider the minimization of the expectation of piecewise (not necessarily convex) quadratic function over Wasserstein balls. This expectation problem often appears as a key sub-problem of distributionally robust optimization problems. We present a computationally accessible semidefinite program (SDP)-based characterization for the optimal distribution of this problem, requiring only the optimal solution of a single SDP. We show that strong duality holds between the expectation problem and its SDP dual problem. We then provide a constructive characterization of the associated optimal distributions and prove that they can be explicitly recovered from an optimal solution of the dual of the dual SDP. This result enables the direct computation of the optimal distributions of the expectation problem. Furthermore, we demonstrate through a numerical study on a distributionally robust mean-risk portfolio optimization problem using simulated data that the worst-case distributions can be computed efficiently and utilized to obtain probabilistic interpretation of worst-case solutions.
N. Dizon, V. Jeyakumar· Optimization Letters· 0 citations
We develop a certified, scalable approximation for high-dimensional Wasserstein distributionally robust portfolio optimization. For expected-utility maximization under order-one Wasserstein ambiguity, standard duality yields a semi-infinite convex program. For long-only portfolios with box support under the one-norm ground metric, an exact sample-specific vertex reformulation provides an exponential-size computational benchmark. We then majorize the utility by supporting hyperplanes and dualize the support subproblems, obtaining a finite hyperplane--dual formulation over compact polyhedral supports. Under the one-norm ground metric and polyhedral portfolio constraints, this formulation is a polynomial-size linear program. The uniform utility-approximation error bounds both the robust-value error and the near-optimality gap for the original robust problem. Experiments validate the certified approximation and demonstrate monthly 476-asset rebalancing and computational scalability to 1,000 assets.
We investigate stochastic simple bilevel optimization with smooth and possibly nonconvex upper- and lower-level objectives. Existing stochastic extensions of dynamic barrier gradient descent (DBGD) either obtain fast convergence under an unverifiable trajectory-dependent ``rare-visit''assumption, or remove this assumption at a substantially higher oracle cost. We show that a simple denominator-only regularization of the DBGD multiplier eliminates the need for such an assumption while preserving fast convergence rates. Specifically, our method achieves $(\varepsilon, \varepsilon)$-stationarity in $O(\varepsilon^{-2})$ iterations using $O(\varepsilon^{-4})$ upper-level and $O(\varepsilon^{-7})$ lower-level stochastic gradients, which improves upon the best assumption-free complexities. We additionally derive anytime parameter schedules.
Daniel Cortild, Mathias Staudigl, J. Peypouquet et al.· 0 citations
We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p<q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include \(\widetilde O(1/T)\) for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $X$ and the set of possible subgradients $H$ are convex, centrally symmetric, and admit a type of minmax theorem, advancing on a fundamental question by Sridharan [Sri12, Section 10.1.2, Q3]. As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.
David Martínez-Rubio, Brian Bullins, Cristóbal Guzmán et al.· 1 citation
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