This work studies how a generalized error bound (GEB) property in the ambient Euclidean space transfers to the active manifold under \(\mathcal{C}^1\) partial smoothness. The GEB generalizes the standard error bound, which upper bounds the distance to a subset of critical points using first-order stationarity residuals, by allowing the distance to be composed with a … Read more
We revisit the framework of generalized Benders decomposition over a compact but non-finite master domain. We show by counterexample that strong general dual functions alone may fail to guarantee convergence. We then define a condition of uniform local strongness and prove that strong general dual functions satisfying this condition guarantee finite \(\epsilon\)-termination. Finally, we show … Read more
In this paper, we propose new adaptive proximal gradient algorithms to solve multiobjective optimization problems, where each objective function is the sum of a differentiable function and a proper, closed, convex function. Utilizing the local behavior of the differentiable terms we propose new adaptive ways to select stepsizes used in proximal gradient scheme. In particular, … Read more
We extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is, for the first time, the ability to consider more than two sets, a task which has been in the mind of researchers for many years … Read more
We consider the probabilistic traveling salesman problem (PTSP) in which customer presences are Bernoulli random variables, and the objective is to determine an a priori tour minimizing the expected traveling cost of the a posteriori tour obtained by skipping absent customers after customer presence is revealed. The existing literature has established that, given an a … Read more
We propose a proximal-type algorithm for nonsmooth and nonconvex optimization problems with composite constraints. The constraint is defined by the composition of a locally upper-\(C^2\) outer function with a locally Lipschitz continuous inner mapping. The method is based on an improvement function that balances objective decrease and constraint satisfaction, and on a surrogate model obtained … 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
Consider a discrete optimal transport problem that has at least two consumers. We show that the Monge and Kantorovich versions of such a discrete optimal transport problem are equivalent for all cost functions if and only if the supply from all the suppliers are equal, and the demand from every consumer is an integral multiple … Read more
The symmetry measure of a convex body \(S\subset\mathbb{R}^n\) is given by: \(\mathrm{sym}(S):=\max\{\alpha\ge0:\ \mathrm{there\ exists}\ x\in S\ \mathrm{such\ that}\ -\alpha(S-x)\subseteq S-x\}\), where such an \(x\) is called a Minkowski center. We prove that every convex body \(S\) admits a \(\sqrt{\frac{n}{\mathrm{sym}(S)}}\)-rounding of \(S\), namely, there exists an origin-centered ellipsoid \(E\) and a center \(c\) such that the … Read more
In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater’s condition), a setting in which existing results are limited. Our main technical contribution is the introduction of radial-type H\”{o}lder error bounds, where the error bound constant depends explicitly on the norm of the … Read more