Jianqiang Cheng

A second-order cone programming approach for linear programs with joint probabilistic constraints

Abstract This paper deals with a special case of Linear Programs with joint Probabilistic Constraints (LPPC) with normally distributed coefficients and independent matrix vector rows. Through the piecewise linear approximation and the piecewise tangent approximation, we approximate the stochastic linear programs with two second-order cone programming (SOCP for short) problems. Furthermore, the optimal values of the two SOCP problems are a lower and upper bound of the original problem respectively. Finally, numerical experiments are given on randomly generated data.

Distributionally Robust Stochastic Knapsack Problem

This paper considers a distributionally robust version of a quadratic knapsack problem. In this model, a subsets of items is selected to maximizes the total profit while requiring that a set of knapsack constraints be satisfied with high probability. In contrast to the stochastic programming version of this problem, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support, and possibly an independence assumption. As for the binary constraints, special interest is given to the corresponding semidefinite programming (SDP) relaxation. While in the case that the model only has a single knapsack constraint we present an SDP reformulation for this relaxation, the case of multiple knapsack constraints is more challenging. Instead, two tractable methods are presented for providing upper and lower bounds (with its associated conservative solution) on the SDP relaxation. An extensive computational study is given to illustrate...

Maximum probability shortest path problem

The maximum probability shortest path problem involves the constrained shortest path problem in a given graph where the arcs resources are independent normally distributed random variables. We maximize the probability that all resource constraints are jointly satisfied while the path cost does not exceed a given threshold. We use a second-order cone programming approximation for solving the continuous relaxation problem. In order to solve this stochastic combinatorial problem, a branch-and-bound algorithm is proposed, and numerical examples on randomly generated instances are given.

Distributionally robust stochastic shortest path problem

Abstract This paper considers a stochastic version of the shortest path problem, the Distributionally Robust Stochastic Shortest Path Problem(DRSSPP) on directed graphs. In this model, the arc costs are deterministic, while each arc has a random delay. The mean vector and the second-moment matrix of the uncertain data are assumed known, but the exact information of the distribution is unknown. A penalty occurs when the given delay constraint is not satisfied. The objective is to minimize the sum of the path cost and the expected path delay penalty. As it is NP-hard, we approximate the DRSSPP with a semidefinite programming (SDP for short) problem, which is solvable in polynomial time and provides tight lower bounds.

Random-payoff two-person zero-sum game with joint chance constraints

We study a two-person zero-sum game where the payoff matrix entries are random and the constraints are satisfied jointly with a given probability. We prove that for the general random-payoff zero-sum game there exists a “weak duality” between the two formulations, i.e., the optimal value of the minimizing player is an upper bound of the one of the maximizing player. Under certain assumptions, we show that there also exists a “strong duality” where their optimal values are equal. Moreover, we develop two approximation methods to solve the game problem when the payoff matrix entries are independent and normally distributed. Finally, numerical examples are given to illustrate the performances of the proposed approaches.

Chance constrained 0–1 quadratic programs using copulas

In this paper, we study 0–1 quadratic programs with joint probabilistic constraints. The row vectors of the constraint matrix are assumed to be normally distributed but are not supposed to be independent. We propose a mixed integer linear reformulation and provide an efficient semidefinite relaxation of the original problem. The dependence of the random vectors is handled by the means of copulas. Finally, numerical experiments are conducted to show the strength of our approach.

A second-order cone programming approximation to joint chance-constrained linear programs

We study stochastic linear programs with joint chance constraints, where the random matrix is a special triangular matrix and the random data are assumed to be normally distributed. The problem can be approximated by another stochastic program, whose optimal value is an upper bound of the original problem. The latter stochastic program can be approximated by two second-order cone programming (SOCP) problems [5]. Furthermore, in some cases, the optimal values of the two SOCPs problems provide a lower bound and an upper bound of the approximated stochastic program respectively. Finally, numerical examples with probabilistic lot-sizing problems are given to illustrate the effectiveness of the two approximations.

A completely positive representation of 0-1 linear programs with joint probabilistic constraints

In this paper, we study 0-1 linear programs with joint probabilistic constraints. The constraint matrix vector rows are assumed to be independent, and the coefficients to be normally distributed. Our main results show that this non-convex problem can be approximated by a convex completely positive problem. Moreover, we show that the optimal values of the latter converge to the optimal values of the original problem. Examples randomly generated highlight the efficiency of our approach.

A chance constrained approach for uplink wireless OFDMA networks

In this paper, we compare individual and joint probabilistic constraints for a resource allocation problem in an uplink (UL) wireless OFDMA network. For this purpose, we formulate the problem as a stochastic linear programming (SLP) problem. Then, we transform this model into equivalent deterministic Second-Order Cone Programming (SOCP) problems. All models are intended to maximize the bit rates throughput of the network subject to subcarrier and power user constraints. Our preliminary numerical results show that the joint chance constraint formulation is slightly conservative than the individual probabilistic one. Finally, we show that our approximation of the deterministic joint probabilistic model is very tight.

Stochastic nonlinear resource allocation problem

Abstract In this paper, we deal with a resource allocation problem modeled as special case of 0-1 Quadratic Programs with joint probabilistic rectangular constraints (QPJPC) with normally distributed coefficients and independent matrix vector rows. We reformulate this problem as a completely positive problem. In addition, the optimal value of the latter problem converges to the optimal value of the original problem under certain conditions. Numerical experiments on randomly generated data are given.