Mixed integer predictive control and shortest path reformulation

Mixed integer predictive control deals with optimizing integer and real control variables over a receding horizon. The mixed integer nature of controls might be a cause of intractability for instances of larger dimensions. To tackle this little issue, we propose a decomposition method which turns the original $n$-dimensional problem into $n$ indipendent scalar problems of lot sizing form. Each scalar problem is then reformulated as a shortest path one and solved through linear programming over a receding horizon. This last reformulation step mirrors a standard procedure in mixed integer programming. The approximation introduced by the decomposition can be lowered if we operate in accordance with the predictive control technique: i) optimize controls over the horizon ii) apply the first control iii) provide measurement updates of other states and repeat the procedure

Parametric programming technique for global optimization of wastewater treatment systems

This paper presents a parametric programming technique for the optimal design of industrial wastewater treatment networks (WTN) featuring multiple contaminants. Inspired in scientific notation and powers of ten, the proposed approach avoids the non-convex bilinear terms through a piecewise decomposition scheme that combines the generation of artificial flowrate variables with a multi-parameterization of the outlet concentration variables. The general non-linear problem (NLP) formulation is replaced by a mixed-integer linear programming (MILP) model that is able to generate near optimal solutions, fast. The performance of the new approach is compared to that of global optimization solver BARON through the solution a few test cases

A Decomposition Method for Mixed-Integer Linear Programming Problems with Angular Structure

An algorithm is presented for solving mixed-integer linear programming problems with an angular structure, based on the decomposition technique of Dantzig and Wolfe. The subproblem is a mixed-integer problem of a smaller size than that of the original one. A sufficient condition for optimality is obtained. In the case where the optimality condition is not satisfied, a search for improving the solution is being continued within a restricted extent. By examining illustrative examples, it is observed that the present algorithm is efficient because it has less computing time than the conventional branch and bound method

On the Structure of Decision Diagram-Representable Mixed Integer Programs with Application to Unit Commitment

Over the past decade, decision diagrams (DDs) have been used to model and solve integer programming and combinatorial optimization problems. Despite successful performance of DDs in solving various discrete optimization problems, their extension to model mixed integer programs (MIPs) such as those appearing in energy applications has been lacking. More broadly, the question which problem structures admit a DD representation is still open in the DDs community. In this paper, we address this question by introducing a geometric decomposition framework based on rectangular formations that provides both necessary and sufficient conditions for a general MIP to be representable by DDs. As a special case, we show that any bounded mixed integer linear program admits a DD representation through a specialized Benders decomposition technique. The resulting DD encodes both integer and continuous variables, and therefore is amenable to the addition of feasibility and optimality cuts through refinement procedures. As an application for this framework, we develop a novel solution methodology for the unit commitment problem (UCP) in the wholesale electricity market. Computational experiments conducted on a stochastic variant of the UCP show a significant improvement of the solution time for the proposed method when compared to the outcome of modern solvers

An algorithm for the global resolution of linear stochastic bilevel programs

The aim of this thesis is to find a technique that allows for the use of decomposition methods known from stochastic programming in the framework of linear stochastic bilevel problems. The uncertainty is modeled as a discrete, finite distribution on some probability space. Two approaches are made, one using the optimal value function of the lower level, whereas the second technique uses the Karush-Kuhn-Tucker conditions of the lower level. Using the latter approach, an integer-programming based algorithm for the global resolution of these problems is presented and evaluated

Mechanism Design via Dantzig-Wolfe Decomposition

In random allocation rules, typically first an optimal fractional point is calculated via solving a linear program. The calculated point represents a fractional assignment of objects or more generally packages of objects to agents. In order to implement an expected assignment, the mechanism designer must decompose the fractional point into integer solutions, each satisfying underlying constraints. The resulting convex combination can then be viewed as a probability distribution over feasible assignments out of which a random assignment can be sampled. This approach has been successfully employed in combinatorial optimization as well as mechanism design with or without money. In this paper, we show that both finding the optimal fractional point as well as its decomposition into integer solutions can be done at once. We propose an appropriate linear program which provides the desired solution. We show that the linear program can be solved via Dantzig-Wolfe decomposition. Dantzig-Wolfe decomposition is a direct implementation of the revised simplex method which is well known to be highly efficient in practice. We also show how to use the Benders decomposition as an alternative method to solve the problem. The proposed method can also find a decomposition into integer solutions when the fractional point is readily present perhaps as an outcome of other algorithms rather than linear programming. The resulting convex decomposition in this case is tight in terms of the number of integer points according to the Carath{\'e}odory's theorem