The \emph{optimal value function} is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the \emph{cost/revenue} generated while \emph{minimizing/maximizing} a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity … Read more
An algorithm framework is proposed for minimizing nonsmooth functions. The framework is variable-metric in that, in each iteration, a step is computed using a symmetric positive definite matrix whose value is updated as in a quasi-Newton scheme. However, unlike previously proposed variable-metric algorithms for minimizing nonsmooth functions, the framework exploits self-correcting properties made possible through … Read more
Resource allocation models in contingency planning aim to mitigate unexpected failures in supply chains due to disruptions with rare occurrence but disastrous consequences. This paper formulates this problems as a two-stage stochastic optimization with a risk-averse recourse function, and proposes a novel computationally tractable solution approach. The method relies on an inexact bundle method and … Read more
Machine learning with big data often involves large optimization models. For distributed optimization over a cluster of machines, frequent communication and synchronization of all model parameters (optimization variables) can be very costly. A promising solution is to use parameter servers to store different subsets of the model parameters, and update them asynchronously at different machines … Read more
Large-scale constrained convex optimization problems arise in several application domains. First-order methods are good candidates to tackle such problems due to their low iteration complexity and memory requirement. The level-set framework extends the applicability of first-order methods to tackle problems with complicated convex objectives and constraint sets. Current methods based on this framework either rely … Read more
We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be inde nite which plays … Read more
The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter $M_f$. However, in many settings the non-differentiable convex function $f(\cdot)$ is not uniformly Lipschitz continuous — for example (i) the classical support vector machine (SVM) problem, (ii) the … Read more
The accelerated gradient algorithm is known to have non-monotonic, periodic convergence behavior in the high momentum regime. If important function parameters like the condition number are known, the momentum can be adjusted to get linear convergence. Unfortunately these parameters are usually not accessible, so instead heuristics are used for deciding when to restart. One of … Read more
In a recent note [8], the author provides a counterexample to the global convergence of what his work refers to as “the DSOS and SDSOS hierarchies” for polynomial optimization problems (POPs) and purports that this refutes claims in our extended abstract [4] and slides in [3]. The goal of this paper is to clarify that … Read more
We develop an algorithm for minimax problems that arise in robust optimization in the absence of objective function derivatives. The algorithm utilizes an extension of methods for inexact outer approximation in sampling a potentially infinite-cardinality uncertainty set. Clarke stationarity of the algorithm output is established alongside desirable features of the model-based trust-region subproblems encountered. We … Read more