John C. Plummer

Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization

The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables. It uses OptQuest, a commercial implementation of scatter search developed by OptTek Systems, Inc., to provide starting points for any gradient-based local solver for nonlinear programming (NLP) problems. This solver seeks a local solution from a subset of these points, holding discrete variables fixed. The procedure is motivated by our desire to combine the superior accuracy and feasibility-seeking behavior of gradient-based local NLP solvers with the global optimization abilities of OptQuest. Computational results include 155 smooth NLP and mixed integer nonlinear program (MINLP) problems due to Floudas et al. (1999), most with both linear and nonlinear constraints, coded in the GAMS modeling language. Some are quite large for global optimization, with over 100 variables and 100 constraints. Global solutions to almost all problems are found in a small number of local solver calls, often one or two.

A bilevel programming approach to determining tax credits for biofuel production

Abstract This paper presents a bilevel programming formulation of a leader–follower game that can be used to help decision makers arrive at a rational policy for encouraging biofuel production. In the model, the government is the leader and would like to minimize the annual tax credits it allows the petro-chemical industry for producing biofuels. The crops grown for this purpose are on land now set aside and subsidized through a different support program. The agricultural sector is the follower. Its objective is to maximize profits by selecting the best mix of crops to grow as well as the percentage of land to set aside. Two solution algorithms are developed. The first involves a grid search over the tax credit variables corresponding to the two biofuels under consideration, ester and ethanol. Once these values are fixed, nonfood crop prices can be determined and the farm sector linear program solved. The second algorithm is based on an approximate nonlinear programming (NLP) formulation of the bilevel program. An “engineering” approach is taken where the discontinuities in the governments problem are ignored and the farm model is treated as a function that maps nonfood crop prices into allocation decisions. Results are given for an agricultural region in the northern part of France comprising 393 farms.

A Multistart Scatter Search Heuristic for Smooth NLP and MINLP Problems

The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables. It uses OptQuest, a commercial implementation of scatter search developed by OptTek Systems, Inc., to provide starting points for a gradient-based local NLP solver. This solver seeks a local solution from a subset of these points, holding discrete variables fixed. The procedure is motivated by our desire to combine the superior accuracy and feasibility-seeking behavior of gradient-based local NLP solvers with the global optimization abilities of OptQuest. Computational results include 144 smooth NLP and MINLP problems due to Floudas et al, most with both linear and nonlinear constraints, coded in the GAMS modeling language. Some are quite large for global optimization, with over 100 variables and many constraints. Global solutions to almost all problems are found in a small number of NLP solver calls, often one or two.

Primal-Dual and Primal Interior Point Algorithms for General Nonlinear Programs

An interior point algorithm for general nonlinear programs is presented. Inequality constraints are converted to equalities with slack variables. All bounds are handled with a barrier term in the objective. The Kuhn-Tucker system of the resulting equality constrained barrier problem is solved directly by Newtons Method. Primal-Dual, Primal, and Primal-Dual with trust region variants are developed and evaluated. An implementation which utilizes the true Lagrangian Hessian and exploits Jacobian and Hessian sparsity is described. Computational results are presented and discussed. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Determining Tax Credits for Converting Nonfood Crops to Biofuels: An Application of Bilevel Programming

This chapter presents two algorithms for solving a bilevel programming problem aimed at deriving optimal tax credits for the production of biofuels. In this problem the government is the leader and would like to minimize the annual subsidy it pays to the petro-chemical industry in the form of tax credits to produce biofuels from crops grown on land now set aside and subsidized through a different support program. The agricultural sector is the follower. Its objective is to maximize profits by selecting the best mix of crops to grow as well as the percentage of land to set aside. The first algorithm involves a grid search over the tax credit variables corresponding to the two biofuels under consideration, ester and ethanol. Once these values are fixed, nonfood crop prices can be determined and the farm sector linear program (LP) solved. The second algorithm is based on an approximate nonlinear programming (NLP) formulation of the bilevel program. An “engineering” approach is taken where the discontinuities in the government’s problem are ignored and the farm model is treated as a function that maps nonfood crop prices into allocation decisions. A standard NLP code called SQP is then employed to solve the problem. Results are given for an agricultural region in the northern part of France comprising 393 farms.

Multistart algorithms for seeking feasibility

This paper describes modifications to two multistart algorithms for global optimization which enable them to find feasible solutions to a system of nonlinear constraints more efficiently. The multistart algorithms, called OptQuest-NLP (OQNLP) and Multistart-NLP (MSNLP), start a local NLP Solver from a set of starting points and return the best solution found. Candidate starting points are generated either by a scatter search heuristic or by a randomized process. Two adaptive filters choose a small subset of the candidate points as starting points. The modifications to facilitate feasibility seeking include replacing the exact penalty function used to measure the goodness of a starting point with the sum of infeasibilities, and terminating when a feasible solution is found. We describe experimental results on a large and diverse set of smooth nonlinear nonconvex problems coded in the GAMS modeling language. These are chosen so that a single application of a selected solver from the user-specified starting point terminates infeasible, yet the problems all have feasible solutions. Our results show that MSNLPs feasibility mode is able to find feasible solutions to almost all problems. It is moderately faster than MSNLP not using feasibility mode, and is somewhat better a finding feasible solutions when they exist. It is now an option within MSNLP, and can be invoked by inserting an appropriate record into the options file.

Optimal design of efficient acoustic antenna arrays

Minimax optimal design of sonar transducer arrays can be formulated as a nonlinear program with many convex quadratic constraints and a nonconvex quadratic efficiency constraint. The variables of this problem are a scaling and phase shift applied to the output of each sensor.This problem is solved by applying Lagrangian relaxation to the convex quadratic constraints.Extensive computational experience shows that this approach can efficiently find near-optimal solutions of problems with up to 391 variables and 579 constraints.

Pseudo-Cut Strategies for Global Optimization

Motivated by the successful use of a pseudo-cut strategy within the setting of constrained nonlinear and nonconvex optimization in Lasdon et al. (2010), we propose a framework for general pseudo-cut strategies in global optimization that provides a broader and more comprehensive range of methods. The fundamental idea is to introduce linear cutting planes that provide temporary, possibly invalid, restrictions on the space of feasible solutions, as proposed in the setting of the tabu search metaheuristic in Glover (1989), in order to guide a solution process toward a global optimum, where the cutting planes can be discarded and replaced by others as the process continues. These strategies can be used separately or in combination, and can also be used to supplement other approaches to nonlinear global optimization. Our strategies also provide mechanisms for generating trial solutions that can be used with or without the temporary enforcement of the pseudo-cuts.

Dynamic filters and randomized drivers for the multi-start global optimization algorithm MSNLP

We present results of extensive computational tests of (i) comparing dynamic filters (first mentioned in an earlier publication addressing a feasibility seeking algorithm) with static filters and (ii) stochastic starting point generators (‘drivers’) for a multi-start global optimization algorithm called MSNLP (Multi-Start Non-Linear Programming). We show how the widely used NLP local solvers CONOPT and SNOPT compare when used in this context. Our computational tests utilize two large and diverse sets of test problems. Best known solutions to most of the problems are obtained competitively, within 30 solver calls, and the best solutions are often located in the first ten calls. The results show that the addition of dynamic filters and new global drivers can contribute to the increased reliability of the MSNLP algorithmic framework.

Solving a large nonlinear programming problem on a vector processing computer

In an earlier paper, the authors formulated an acoustic antenna array design problem as a nonlinear program. Computation times for large problems (about 400 sensors) were in the range of 8 to 10 hours on a Vax 11/780. Most of this time was spent in computing the objective function and its gradient. This paper describes how these computations (and these only) were recoded to exploit the vector processing capabilities of a Cray 1-M computer. Run times are reduced to less than one minute. The results have implications for many nonlinear programs whose function evaluations are very time consuming.