Solutions of mixed-integer second-order cone programming problems for discrete optimization under data uncertainty

Abstract

We consider discrete optimization problems under data uncertainty. Discrete optimization problems, including combinatorial optimization and integer programming problems, are utilized in various fields such as engineering, natural science, and operations research. In recent years, many researchers have attempted to formulate mathematical models using stochastic programming, robust optimization, and distributionally robust optimization, etc., to reflect data uncertainty in the real-world problems. However, not only are many discrete optimization problems difficult to solve themselves, but there are originally tractable ones that become difficult as well when data uncertainty is considered. We represent discrete optimization problems with some specific assumptions about data uncertainty as mixed-integer second-order cone programming problems and present several algorithms for these problems. First, we propose some algorithms to solve a chance-constrained binary knapsack problem with weight uncertainty. The binary knapsack problem aims at efficient resource allocation with a capacity constraint and indivisible items. It can also be subroutines during the procedures of a branch-and-cut and a branch-and-price algorithm when a solution set of a mixed-integer programming problem includes capacity constraints. A chance constraint can replace the capacity constraint if the weights of items are random variables, and it can also be reformulated as the second-order cone constraint under some specific assumptions about the distributions of random weights. As a result, the problem becomes the second-order cone-constrained binary knapsack problem. We propose an algorithm to obtain the upper and lower bounds on its optimal value by approximating the second-order cone constraint through a robust optimization approach. Moreover, we present a pseudo-polynomial time algorithm by showing that the solution providing the upper bound can be an optimal one to the problem if the accuracy of the approximation reaches a theoretically certain level or more. Computational experiments on several types of randomly generated instances are presented to show the efficiency of the algorithms. Next, we develop an exact algorithm for mean-standard deviation combinatorial optimization problems to consider combinatorial optimization problems in general, which have uncertainty in the objective cost coefficients. There are many combinatorial optimization problems that can be solved using polynomial or pseudo-polynomial time algorithms, such as the binary knapsack problem, the shortest path problem, the minimum cut problem, etc. They appear in many real-world problems and are also utilized as subproblems for large-sized optimization problems. We assume that the uncertain coefficients are independent random variables, and the mean and the standard deviation of each random variable are known. We can define $\rho$-quantile, also called value-at-risk(VaR) at level $1-\rho$, as a measure of the risk of loss for the objective function. Under some specific assumptions about the distributions of cost coefficients, the problem of minimizing the risk measure is equivalent to the mean-standard deviation combinatorial optimization problem with the deterministic second-order cone objective function. We propose a fully polynomial-time approximation scheme to derive an approximate solution whose objective function value is at most $(1 + \epsilon)$ times the optimal value of the problem. We also develop an iterative algorithm that solves a number of ordinary combinatorial optimization problems to obtain an optimal solution for the problem. The fitness of the algorithm is also proven. The exact algorithm was tested on the mean-standard deviation binary knapsack problem and the mean-standard deviation shortest path problem. Lastly, we deal with a single-source capacitated facility location problem with demand uncertainty. Specific distributions or an ambiguity set of possible distributions can be designated to describe uncertain demands. The mathematical model can be reformulated as an allocation-based formulation by using Dantzig-Wolfe decomposition, and a branch-and-price algorithm is applied for the problem. We consider the structure of the master problem and the subproblem, the GRASP heuristic for initial columns, and several techniques to improve the performance of the algorithm. Computational experiments show that our branch-and-price algorithm outperforms CPLEX, which solves the quadratically constrained mixed-integer reformulation of the problem.