Dual Bounds for Periodical Stochastic Programs

A construction of the dual of a periodical formulation of infinite-horizon linear stochastic programs with a discount factor is discussed. The dual problem is used for computing a deterministic upper bound for the optimal value of the considered multistage stochastic program. Numerical experiments demonstrate behavior of that upper bound, especially when the discount factor is close to one.

Using ℓ1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA

Dual Bounds of Sparse Principal Component Analysis Sparse principal component analysis (PCA) is a widely used dimensionality reduction tool in machine learning and statistics. Compared with PCA, sparse PCA enhances the interpretability by incorporating a sparsity constraint. However, unlike PCA, conventional heuristics for sparse PCA cannot guarantee the qualities of obtained primal feasible solutions via associated dual bounds in a tractable fashion without underlying statistical assumptions. In “Using L1-Relaxation and Integer Programming to Obtain Dual Bounds for Sparse PCA,” Santanu S. Dey, Rahul Mazumder, and Guanyi Wang present a convex integer programming (IP) framework of sparse PCA to derive dual bounds. They show the worst-case results on the quality of the dual bounds provided by the convex IP. Moreover, the authors empirically illustrate that the proposed convex IP framework outperforms existing sparse PCA methods of finding dual bounds.

Dynamic Ride-Hailing with Electric Vehicles

We consider the problem of an operator controlling a fleet of electric vehicles for use in a ride-hailing service. The operator, seeking to maximize profit, must assign vehicles to requests as they arise as well as recharge and reposition vehicles in anticipation of future requests. To solve this problem, we employ deep reinforcement learning, developing policies whose decision making uses [Formula: see text]-value approximations learned by deep neural networks. We compare these policies against a reoptimization-based policy and against dual bounds on the value of an optimal policy, including the value of an optimal policy with perfect information, which we establish using a Benders-based decomposition. We assess performance on instances derived from real data for the island of Manhattan in New York City. We find that, across instances of varying size, our best policy trained with deep reinforcement learning outperforms the reoptimization approach. We also provide evidence that this policy may be effectively scaled and deployed on larger instances without retraining.

On generalized surrogate duality in mixed-integer nonlinear programming

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global $$\epsilon $$ ϵ -optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments.

Optimistic Monte Carlo Tree Search with Sampled Information Relaxation Dual Bounds

In the paper, “Optimistic Monte Carlo Tree Search with Sampled Information Relaxation Dual Bounds,” the authors propose an extension to Monte Carlo tree search that uses the idea of “sampling the future” to produce noisy upper bounds on nodes in the decision tree. These upper bounds can help guide the tree expansion process and produce decision trees that are deeper rather than wider, in effect concentrating computation toward more useful parts of the state space. The algorithm’s effectiveness is illustrated in a ride-sharing setting, where a driver/vehicle needs to make dynamic decisions regarding trip acceptance and relocations.

Machine Learning-Guided Dual Heuristics and New Lower Bounds for the Refueling and Maintenance Planning Problem of Nuclear Power Plants

This paper studies the hybridization of Mixed Integer Programming (MIP) with dual heuristics and machine learning techniques, to provide dual bounds for a large scale optimization problem from an industrial application. The case study is the EURO/ROADEF Challenge 2010, to optimize the refueling and maintenance planning of nuclear power plants. Several MIP relaxations are presented to provide dual bounds computing smaller MIPs than the original problem. It is proven how to get dual bounds with scenario decomposition in the different 2-stage programming MILP formulations, with a selection of scenario guided by machine learning techniques. Several sets of dual bounds are computable, improving significantly the former best dual bounds of the literature and justifying the quality of the best primal solution known.

Improving Lagrange Dual Bounds for Quadratic Extremal Problems

Introduction. Due to the fact that quadratic extremal problems are generally NP-hard, various convex relaxations to find bounds for their global extrema are used, namely, Lagrangian relaxation, SDP-relaxation, SOCP-relaxation, LP-relaxation, and others. This article investigates a dual bound that results from the Lagrangian relaxation of all constraints of quadratic extremal problem. The main issue when using this approach for solving quadratic extremal problems is the quality of the obtained bounds (the magnitude of the duality gap) and the possibility to improve them. While for quadratic convex optimization problems such bounds are exact, in other cases this issue is rather complicated. In non-convex cases, to improve the dual bounds (to reduce the duality gap) the techniques, based on ambiguity of the problem formulation, can be used. The most common of these techniques is an extension of the original quadratic formulation of the problem by introducing the so-called functionally superfluous constraints (additional constraints that result from available constraints). The ways to construct such constraints can be general in nature or they can use specific features of the concrete problems. The purpose of the article is to propose methods for improving the Lagrange dual bounds for quadratic extremal problems by using technique of functionally superfluous constraints; to present examples of constructing such constraints. Results. The general concept of using functionally superfluous constraints for improving the Lagrange dual bounds for quadratic extremal problems is considered. Methods of constructing such constraints are presented. In particular, the method proposed by N.Z. Shor for constructing functionally superfluous constraints for quadratic problems of general form is presented in generalized and schematized forms. Also it is pointed out that other special techniques, which employ the features of specific problems for constructing functionally superfluous constraints, can be used. Conclusions. In order to improve dual bounds for quadratic extremal problems, one can use various families of functionally superfluous constraints, both of general and specific type. In some cases, their application can improve bounds or even provide an opportunity to obtain exact values of global extrema.