Abstract
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 epsilon -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.
Highlights
In this paper we consider problems that can be expressed in the form min f (x) + πi g(xi, ci ), x∈X i ∈I (1)where g(xi, ci ) is the optimal solution of the LP subproblem SP : g(xi, ci ) := min yi ∈Y {ci C yi | Ayi ≤ Bxi }
This paper presents a novel concept of inexact oracles that can be used to speed up the computational time of Benders-type decomposition algorithms for a class of large scale optimization problems
We propose two adaptive oracles that yield inexact and progressively more accurate information when subproblems are not solved
Summary
In this paper we consider problems that can be expressed in the form min f (x) + πi g(xi , ci ), x∈X i ∈I (1). The coefficient matrices A, B, and C are the same in every subproblem so the subproblems differ only through the value of their parameters, xi and ci. The function g(xi , ci ) has properties that can be exploited in the solution procedure It is a saddle function, convex w.r.t. xi , and concave w.r.t. ci. The investments affect a set I of situations, and xi is the subvector of x that represents the investments that affect the situation i, ci specifies the operational costs, yi defines the operational decisions, and g(xi , ci ) gives the optimal operating cost. The test case we present in this paper is a stochastic problem, where the situations are the different possible scenarios that might occur in the future, each of which is weighted by the probability πi of that situation occurring. Note that in a multistage planning problem the sum of the πi at each stage is equal to 1, so i∈I πi ≥ 1
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