Abstract
The accelerated prox-level (APL) and uniform smoothing level (USL) methods recently proposed by Lan (Math Program, 149: 1–45, 2015) can achieve uniformly optimal complexity when solving black-box convex programming (CP) and structure non-smooth CP problems. In this paper, we propose two modified accelerated bundle-level type methods, namely, the modified APL (MAPL) and modified USL (MUSL) methods. Compared with the original APL and USL methods, the MAPL and MUSL methods reduce the number of subproblems by one in each iteration, thereby improving the efficiency of the algorithms. Conclusions of optimal iteration complexity of the proposed algorithms are established. Furthermore, the modified methods are applied to the two-stage stochastic programming, and numerical experiments are implemented to illustrate the advantages of our methods in terms of efficiency and accuracy.
Highlights
In the fields of production planning, finance risk, telecommunication, and electricity, decision makers need to take into consideration uncertainty about the information and the model itself.Lack of data, calculation errors as well as unpredictability, etc. lead to the uncertainty in information.The uncertainty of model is derived from the structure of problem, the features of constraints, as well as the risks and profiles of decisions
The modified methods are applied to the two-stage stochastic programming, and numerical experiments are implemented to illustrate the advantages of our methods in terms of efficiency and accuracy
We proposed the modified accelerated prox-level (MAPL) method which requires only one subproblem to be solved per iteration while achieving the uniformly optimal iteration complexity for solving the black-box convex programming (CP) problems
Summary
In the fields of production planning, finance risk, telecommunication, and electricity, decision makers need to take into consideration uncertainty about the information and the model itself. Through a practical problem, network planning with random demand, we introduce the mathematical model of two-stage stochastic programming. As an extension of the network planning problem, the standard mathematical model of two-stage stochastic programming with recourse is given, which is more convenient for generalization and theoretical analysis. The two-stage stochastic programming with fixed recourse is in the following form min s.t. where x is the decision variable, c is the the cost of production and Q( x, D ) denotes the optimal objective value of the second-stage problem min s.t. Wy = h − Tx, y ≥ 0. We apply the proposed methods to solve the two-stage stochastic programming problems with recourse.
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