MPC

MPC 2021, ISSUE 4



Mathematical Programming Computation, Volume 12, Issue 4, December 2021

A Benders squared (B^2) framework for infinite-horizon stochastic linear programs

Giacomo Nannicini, Emiliano Traversi, Roberto Wolfler Calvo

We propose a nested decomposition scheme for infinite-horizon stochastic linear programs. Our approach can be seen as a provably convergent extension of stochastic dual dynamic programming to the infinite-horizon setting: we explore a sequence of finite-horizon problems of increasing length until we can guarantee convergence with a given confidence level. The methodology alternates between a forward pass to explore sample paths and determine trial solutions, and a backward pass to generate a polyhedral approximation of the optimal value function by computing subgradients from the dual of the scenario subproblems. A computational study on a large set of randomly generated instances for two classes of problems shows that the proposed algorithm is able to effectively solve instances of moderate size to high precision, provided that the instance structure allows the construction of what we call constant-statepolicies with satisfactory objective function value.

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Mathematical Programming Computation, Volume 12, Issue 4, December 2021

Benders decomposition with adaptive oracles for large scale optimization

Nicolò Mazzi, Andreas Grothey, Ken McKinnon, Nagisa Sugishita

This paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an ϵ-optimal solution: an illustrative case is 31.9 times faster for a 1.00% convergence tolerance and 15.4 times faster for a 0.01% tolerance.

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Mathematical Programming Computation, Volume 12, Issue 4, December 2021

A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs

Rohit Kannan, James R. Luedtke

We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraints violation. To this end, we construct a reformulated problem whose objective is to minimize the probability of constraints violation subject to deterministic convex constraints (which includes a bound on the objective function value). We adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem, and apply a projected stochastic subgradient algorithm to solve it. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to stationary solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Our proposal also includes a tailored implementation of the smoothing-based approach that chooses key algorithmic parameters based on problem data. Computational results on four test problems from the literature indicate that our proposed approach can efficiently determine good approximations of the efficient frontier.

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Mathematical Programming Computation, Volume 12, Issue 4, December 2021

Computational aspects of infeasibility analysis in mixed integer programming

Jakob Witzig, Timo Berthold, Stefan Heinz

The analysis of infeasible subproblems plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. There are two fundamentally different concepts to generate valid global constraints from infeasible subproblems: conflict graph analysis and dual proof analysis. While conflict graph analysis detects sets of contradicting variable bounds in an implication graph, dual proof analysis derives valid linear constraints from the proof of the dual LP’s unboundedness. The main contribution of this paper is twofold. Firstly, we present three enhancements of dual proof analysis: presolving via variable cancellation, strengthening by applying mixed integer rounding functions, and a filtering mechanism. Further, we provide a comprehensive computational study evaluating the impact of every presented component regarding dual proof analysis. Secondly, this paper presents the first combined approach that uses both conflict graph and dual proof analysis simultaneously within a single MIP solution process. All experiments are carried out on general MIP instances from the standard public test set Miplib 2017; the presented algorithms have been implemented within the non-commercial MIP solver SCIP and the commercial MIP solver FICO Xpress.

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