Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

Toward global optimization of neural networks: a comparison of the genetic algorithm and backpropagation

Abstract: The recent surge in activity of neural network research in business is not surprising since the underlying functions controlling business data are generally unknown and the neural network offers a tool that can approximate the unknown function to any degree of desired accuracy. The vast majority of these studies rely on a gradient algorithm, typically a variation of backpropagation, to obtain the parameters (weights) of the model. The well-known limitations of gradient search techniques applied to complex nonlinear optimization problems such as artificial neural networks have often resulted in inconsistent and unpredictable performance. Many researchers have attempted to address the problems associated with the training algorithm by imposing constraints on the search space or by restructuring the architecture of the neural network. In this paper we demonstrate that such constraints and restructuring are unnecessary if a sufficiently complex initial architecture and an appropriate global search algorithm is used. We further show that the genetic algorithm cannot only serve as a global search algorithm but by appropriately defining the objective function it can simultaneously achieve a parsimonious architecture. The value of using the genetic algorithm over backpropagation for neural network optimization is illustrated through a Monte Carlo study which compares each algorithm on in-sample, interpolation, and extrapolation data for seven test functions.

Perspective cuts for a class of convex 0–1 mixed integer programs

Abstract: We show that the convex envelope of the objective function of Mixed-Integer Programming problems with a specific structure is the perspective function of the continuous part of the objective function. Using a characterization of the subdifferential of the perspective function, we derive “perspective cuts”, a family of valid inequalities for the problem. Perspective cuts can be shown to belong to the general family of disjunctive cuts, but they do not require the solution of a potentially costly nonlinear programming problem to be separated. Using perspective cuts substantially improves the performance of Branch & Cut approaches for at least two models that, either “naturally” or after a proper reformulation, have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.

On characterizing the “knee” of the Pareto curve based on Normal-Boundary Intersection

Abstract: This paper deals with the issue of generating one Pareto optimal point that is guaranteed to be in a “desirable” part of the Pareto set in a given multicriteria optimization problem. A parameterization of the Pareto set based on the recently developed normal-boundary intersection technique is used to formulate a subproblem, the solution of which yields the point of “maximum bulge”, often referred to as the “knee of the Pareto curve”. This enables the identification of the “good region” of the Pareto set by solving one nonlinear programming problem, thereby bypassing the need to generate many Pareto points. Further, this representation extends the concept of the “knee” for problems with more than two objectives. It is further proved that this knee is invariant with respect to the scales of the multiple objective functions. The generation of this knee however requires the value of each objective function at the minimizer of every objective function (the pay-off matrix). The paper characterizes situations when approximations to the function values comprising the pay-off matrix would suffice in generating a good approximation to the knee. Numerical results are provided to illustrate this point. Further, a weighted sum minimization problem is developed based on the information in the pay-off matrix, by solving which the knee can be obtained.

Optimization of Large-Scale Hydropower System Operations

Abstract: A practical monthly optimization model, called SISOPT, is developed for the management and operations of the Brazilian hydropower system. The system, one of the largest in the world, consists of 75 hydropower plants with an installed capacity of 69,375 MW, producing 92% of the nation's electrical power. The system size and nonlinearity pose a real challenge to the modelers. The basic model is formulated in nonlinear programming ~NLP!. The NLP model is the most general formulation and provides a foundation for analysis by other methods. The formulated NLP model was first linearized by two different linearization techniques and solved by linear programming ~LP!. A comparative analysis was made of the results obtained from the linearized and the NLP models. The results show that the simplest linearized model ~referred to as the LP model! without iteration is suitable for planning purposes. For example, the LP model could be used in system capacity expansion studies or to explore various design parameters in connection with feasibility studies, where details in storage variation are not as important as the power production. With a good initial policy provided by the LP model, the successive linear programming ~SLP! model produced excellent results with fast convergence. The NLP model, the most complex and accurate model in the suite, is particularly suited for setting up guidelines for real-time operations using inflow forecast with frequent updating. The performance of the NLP model was checked against the historical operational records, and the comparison yields indica- tions of superior performance.

Non-Linear Programming—Theory and Methods

Abstract: (1977). Non-Linear Programming—Theory and Methods. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 4, pp. 895-895.