Abstract
We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.
Highlights
Mathematical programming is an invaluable tool for optimal decision-making that was initially developed in a deterministic setting
We remark that our formulation of the hydro scheduling problem is an instance of (CCP) rather than the more general (CCPR) because we do not take into account recovery costs, but this is allowed by the algorithm that we propose, by dropping constraints and auxiliary variables related to φ(x, wi ), Cx in (CCPR)
In our implementation we considered the following alternatives to determine how and when to perform separation: sepAll Separation is performed at integer-feasible solutions in the Branch-and-Cut search for each scenario i having associated variable zi = 0; sepGroup Scenarios are partitioned in subsets, where each subset includes those scenarios of the scenario tree having a common ancestor at the second time period
Summary
Mathematical programming is an invaluable tool for optimal decision-making that was initially developed in a deterministic setting. We consider a problem formulation with step price functions that involves binary variables in the sets Cx (wi ), and apply the Branch-and-Cut algorithm to solve the continuous relaxation and to generate primal bounds as a heuristic. We remark that our formulation of the hydro scheduling problem is an instance of (CCP) rather than the more general (CCPR) because we do not take into account recovery costs, but this is allowed by the algorithm that we propose, by dropping constraints and auxiliary variables related to φ(x, wi ), Cx (wi ) in (CCPR). 1 how to obtain a deterministic equivalent formulation for (CCPR) using binary variables We introduce this mathematical model for the linear case, to explain the basic ideas and notation before transitioning to the nonlinear convex case, which is the focus of this paper. Assuming the functions gij , gij are convex, (2) is a convex MINLP in the sense that it has a convex continuous relaxation
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