In this paper, we propose a new multilevel stochastic framework for the solution of optimization problems. The proposed approach uses random regularized first-order models that exploit an available hierarchical description of the problem, being either in the classical variable space or in the function space, meaning that different levels of accuracy for the objective function … Read more
This paper studies effective scenarios in Distributionally Robust Optimization (DRO) problems defined on a finite number of realizations (also called scenarios) of the uncertain parameters. Effective scenarios are critical scenarios in DRO in the sense that their removal from the support of the considered distributions alters the optimal value. Ineffective scenarios are those whose removal … Read more
The scenario approach is widely used in robust control system design and chance-constrained optimization, maintaining convexity without requiring assumptions about the probability distribution of uncertain parameters. However, the approach can demand large sample sizes, making it intractable for safety-critical applications that require very low levels of constraint violation. To address this challenge, we propose a … Read more
In this paper, we introduce an approach for obtaining probabilistically guaranteed upper and lower bounds on the true optimal value of stopping problems. Bounds of existing simulation-and-regression approaches, such as those based on least squares Monte Carlo and information relaxation, are stochastic in nature and therefore do not come with a finite sample guarantee. Our … Read more
Distributionally robust optimization (DRO) studies decision problems under uncertainty where the probability distribution governing the uncertain problem parameters is itself uncertain. A key component of any DRO model is its ambiguity set, that is, a family of probability distributions consistent with any available structural or statistical information. DRO seeks decisions that perform best under the … Read more
In this paper, we revisit the recent stepsize applied for the gradient descent scheme which is called NGD proposed by [Hoai et al., A novel stepsize for gradient descent method, Operations Research Letters (2024) 53, doi: 10.1016/j.orl.2024.107072]. We first investigate NGD stepsize with two well-known accelerated techniques which are Heavy ball and Nesterov’s methods. In … Read more
Sample average approximation (SAA) is a technique for obtaining approximate solutions to stochastic programs that uses the average from a random sample to approximate the expected value that is being optimized. Since the outcome from solving an SAA is random, statistical estimates on the optimal value of the true problem can be obtained by solving … Read more
The exponential growth in data availability in recent years has led to new formulations of data-driven optimization problems. One such formulation is that of stochastic optimization problems with contextual information, where the goal is to optimize the expected value of a certain function given some contextual information (also called features) that accompany the main data … Read more
We consider improving the polyhedral representation of the extensive form of a stochastic mixed-integer program (SMIP). Given a facet for a single-scenario version of an SMIP, our main result provides necessary and sufficient conditions under which this inequality remains facet-defining for the extensive form. We then present several implications, which show that common recourse structures … Read more
We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. A new family of Lagrangian cuts, termed “ReLU Lagrangian cuts,” is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition methods. We show that including ReLU Lagrangian cuts is sufficient to … Read more