Realistic decision problems are inherently multiobjective and uncertain. To manage both of these complexities, robust multiobjective optimization strives to aid the decision maker in finding a decision which is Pareto efficient and is hedged against the worst-case scenario. In this paper, we present a robustness approach which is grounded in the decision maker’s preferences by … Read more
In derivative-free optimization (DFO), one minimizes functions for which the gradient is unavailable or expensive to compute. In many applications, objective function values and gradients are noisy due to simulations or system randomness. A class of standard direct-search methods for DFO accept a trial point when it decreases the objective function by an amount proportional … Read more
Decision scenarios far from rare in practice often place data-driven decision-making in an awkward position, where the decision maker may have access to neither the target distribution itself nor any empirical samples drawn from it, especially when decisions must be made in novel or highly uncertain environments. In response, robust out-of-distribution stochastic optimization (RooDSO) has … Read more
Mixed integer nonlinear programs with binary coupling decisions naturally model selective coordination tasks where a fixed penalty is incurred whenever adjacent continuous variables differ. A prominent example is the classical Potts model, which is widely used in statistical inference. However, exact solvability remains theoretically challenging since the problem is NP hard on general graphs, and … Read more
This document provides an extended and rigorous framework dedicated to the geometric analysis of the 3D incompressible Navier-Stokes equations \cite{ESS2003}. We comprehensively develop geometric proofs related to decay estimates, blow-up profiles, and the foundational partial regularity theory of Caffarelli, Kohn, and Nirenberg (CKN). We examine in detail the Hausdorff dimension of potential singular sets, local … Read more
Self-intersection is a fundamental feasibility constraint in shape optimization: a self-crossing boundary leaves its interior, normal field, and finite-element mesh ill-defined, yet most existing barriers rely on heuristic geometric-proximity measures rather than certifying self-intersection directly. We propose a self-intersection barrier grounded in algebraic detection. We derived two bivariate polynomials from a curve’s Fourier coefficients whose … Read more
In this paper, we study nonconvex stochastic optimization with {decision-dependent distributions}, where the decision variable influences the underlying sampling distribution and only stochastic function-value feedback is available. We address two challenges {induced by decision-dependent distributions}: transport error in momentum-based gradient tracking and variance inflation in zeroth-order estimation. We first develop a Polyak-momentum zeroth-order method that … Read more
We develop a novel Dantzig-Wolfe (DW) decomposition algorithm for monotone two-stage stochastic mixed-integer programs (SMIPs). The key novelty in this algorithm is to relax the non-anticipativity constraints (NACs) in line with the monotonicity in the problem. We prove (i) that the relaxed restricted master problem (RMP) faster identifies dominated columns that cannot improve the RMP … Read more
The ESA/NASA joint LISA (laser interferometer space antenna) mission is designed to detect gravitational waves to perform gravitational astronomy. A key mission requirement is the maintenance of a three-spacecraft constellation in a near-equilateral triangular configuration with a prescribed inter-spacecraft separation. Existing approaches have addressed this problem using simplified dynamical models to enhance tractability; however, the … Read more
We develop a few calculus rules to compute or lower bound the facial distance of a polytope. We illustrate our calculus rules on various popular polytopes. In particular, we provide a lower bound on the facial distance of the Birkhoff polytope. CitationWorking paper. Tepper School of Business. Carnegie Mellon UniversityArticleDownload View PDF