Claudia D’Ambrosio

Mixed integer nonlinear programming tools: a practical overview

We present a review of available tools for solving mixed integer nonlinear programming problems. Our aim is to give the reader a flavor of the difficulties one could face and to discuss the tools one could use to try to overcome such difficulties.

A Storm of Feasibility Pumps for Nonconvex MINLP

One of the foremost difficulties in solving Mixed-Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed-Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed-Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pump algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pump algorithms is that, unlike most previous successive projection methods from the literature, projection is “naturally” taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performance of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.

Experiments with a feasibility pump approach for nonconvex MINLPs

We present a new Feasibility Pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programming problems. Differences with the previously proposed Feasibility Pump algorithms and difficulties arising from nonconvexities in the models are extensively discussed. The main methodological innovations of this variant are: (a) the first subproblem is a nonconvex continuous Nonlinear Program, which is solved using global optimization techniques; (b) the solution method for the second subproblem is complemented by a tabu list. We exhibit computational results showing the good performance of the algorithm on instances taken from the MINLPLib.

Mathematical Programming techniques in Water Network Optimization

In this article we survey mathematical programming approaches to problems in the field of drinking water distribution network optimization. Among the predominant topics treated in the literature, we focus on two different, but related problem classes. One can be described by the notion of network design, while the other is more aptly termed by network operation. The basic underlying model in both cases is a nonlinear network flow model, and we give an overview on the more specific modeling aspects in each case. The overall mathematical model is a Mixed Integer Nonlinear Program having a common structure with respect to how water dynamics in pipes are described. Finally, we survey the algorithmic approaches to solve the proposed problems and we discuss computation on various types of water networks.

Mixed integer nonlinear programming tools: an updated practical overview

We present a review of available tools for solving mixed integer nonlinear programming problems. Our aim is to give the reader a flavor of the difficulties one could face and to discuss the tools one could use to try to overcome such difficulties.

An Algorithmic Framework for MINLP with Separable Non-Convexity

We present an algorithm for Mixed-Integer Nonlinear Programming (MINLP) problems in which the non-convexity in the objective and constraint functions is manifested as the sum of non-convex univariate functions. We employ a lower bounding convex MINLP relaxation obtained by approximating each non-convex function with a piecewise-convex underestimator that is repeatedly refined. The algorithm is implemented at the level of a modeling language. Favorable numerical results are presented.

A global-optimization algorithm for mixed-integer nonlinear programs having separable non-convexity

We present a global optimization algorithm for MINLPs (mixed-integer nonlinear programs) where any non-convexity is manifested as sums of non-convex univariate functions. The algorithm is implemented at the level of a modeling language, and we have had substantial success in our preliminary computational experiments.

Box-Constrained Mixed-Integer Polynomial Optimization Using Separable Underestimators

We propose a novel approach to computing lower bounds for box-constrained mixed-integer polynomial minimization problems. Instead of considering convex relaxations, as in most common approaches, we determine a separable underestimator of the polynomial objective function, which can then be minimized easily over the feasible set even without relaxing integrality. The main feature of our approach is the fast computation of a good separable underestimator; this is achieved by computing tight underestimators monomialwise after an appropriate shifting of the entire polynomial. If the total degree of the polynomial objective function is bounded, it suffices to consider finitely many monomials, the optimal underestimators can then be computed offline and hardcoded. For the quartic case, we determine all optimal monomial underestimators analytically.

Controlling the Average Behavior of Business Rules Programs

Business Rules are a programming paradigm for non-programmer business users. They are designed to encode empirical knowledge of a business unit by means of “if-then” constructs. The classic example is that of a bank deciding whether to open a line of credit to a customer, depending on how the customer answers a list of questions. These questions are formulated by bank managers on the basis of the bank strategy and their own experience. Banks often have goals about target percentages of allowed loans. A natural question then arises: can the Business Rules be changed so as to meet that target on average? We tackle the question using “machine learning constrained” mathematical programs, which we solve using standard off-the-shelf solvers. We then generalize this to arbitrary decision problems.

Optimal Scheduling of a Multiunit Hydro Power Station in a Short-Term Planning Horizon

This chapter deals with the problem of determining the commitment and the power generation of a single-reservoir pump storage hydro power plant. Two MILP models with different levels of complexity are computationally tested and compared with the natural MINLP formulation. In this specific optimization problem, the quality of the approximation provided by the piecewise linear approximation of nonlinear and nonconcave constraints is very effective in order to exploit the performance of MILP solvers.