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.

Optimization of Power System Operation

Abstract: Preface. 1 Introduction. 1.1 Conventional Methods. 1.2 Intelligent Search Methods. 1.3 Application of Fuzzy Set Theory. 2 Power Flow Analysis. 2.1 Mathematical Model of Power Flow. 2.2 Newton-Raphson Method. 2.3 Gauss-Seidel Method. 2.4 P-Q decoupling Method. 2.5 DC Power Flow. 3 Sensitivity Calculation. 3.1 Introduction. 3.2 Loss Sensitivity Calculation. 3.3 Calculation of Constrained Shift Sensitivity Factors. 3.4 Perturbation Method for Sensitivity Analysis. 3.5 Voltage Sensitivity Analysis. 3.6 Real-Time Application of Sensitivity Factors. 3.7 Simulation Results. 3.8 Conclusion. 4 Classic Economic Dispatch. 4.1 Introduction. 4.2 Input-Output Characteristic of Generator Units. 4.3 Thermal System Economic Dispatch Neglecting Network Losses. 4.4 Calculation of Incremental Power Losses. 4.5 Thermal System Economic Dispatch with Network Losses. 4.6 Hydrothermal System Economic Dispatch. 4.7 Economic Dispatch by Gradient Method. 4.8 Classic Economic Dispatch by Genetic Algorithm. 4.9 Classic Economic Dispatch by Hopfi eld Neural Network. 5 Security-Constrained Economic Dispatch. 5.1 Introduction. 5.2 Linear Programming Method. 5.3 Quadratic Programming Method. 5.4 Network Flow Programming Method. 5.5 Nonlinear Convex Network Flow Programming Method. 5.6 Two-Stage Economic Dispatch Approach. 5.7 Security-Constrained ED by Genetic Algorithms. 6 Multiarea System Economic Dispatch. 6.1 Introduction. 6.2 Economy of Multiarea Interconnection. 6.3 Wheeling. 6.4 Multiarea Wheeling. 6.5 MAED Solved by Nonlinear Convex Network Flow Programming. 6.6 Nonlinear Optimization Neural Network Approach. 6.7 Total Transfer Capability Computation in Multiareas. 7 Unit Commitment. 7.1 Introduction. 7.2 Priority Method. 7.3 Dynamic Programming Method. 7.4 Lagrange Relaxation Method. 7.5 Evolutionary Programming-Based Tabu Search Method. 7.6 Particle Swarm Optimization for Unit Commitment. 7.7 Analytic Hierarchy Process. 8 Optimal Power Flow. 8.1 Introduction. 8.2 Newton Method. 8.3 Gradient Method. 8.4 Linear Programming OPF. 8.5 Modifi ed Interior Point OPF. 8.6 OPF with Phase Shifter. 8.7 Multiple-Objectives OPF. 8.8 Particle Swarm Optimization for OPF. 9 Steady-State Security Regions. 9.1 Introduction. 9.2 Security Corridors. 9.3 Traditional Expansion Method. 9.4 Enhanced Expansion Method. 9.5 Fuzzy Set and Linear Programming. 10 Reactive Power Optimization. 10.1 Introduction. 10.2 Classic Method for Reactive Power Dispatch. 10.3 Linear Programming Method of VAR Optimization. 10.4 Interior Point Method for VAR Optimization Problem. 10.5 NLONN Approach. 10.6 VAR Optimization by Evolutionary Algorithm. 10.7 VAR Optimization by Particle Swarm Optimization Algorithm. 10.8 Reactive Power Pricing Calculation. 11 Optimal Load Shedding. 11.1 Introduction. 11.2 Conventional Load Shedding. 11.3 Intelligent Load Shedding. 11.4 Formulation of Optimal Load Shedding. 11.5 Optimal Load Shedding with Network Constraints. 11.6 Optimal Load Shedding without Network Constraints. 11.7 Distributed Interruptible Load Shedding. 11.8 Undervoltage Load Shedding. 11.9 Congestion Management. 12 Optimal Reconfi guration of Electrical Distribution Network. 12.1 Introduction. 12.2 Mathematical Model of DNRC. 12.3 Heuristic Methods. 12.4 Rule-Based Comprehensive Approach. 12.5 Mixed-Integer Linear Programming Approach. 12.6 Application of GA to DNRC. 12.7 Multiobjective Evolution Programming to DNRC. 12.8 Genetic Algorithm Based on Matroid Theory. 13 Uncertainty Analysis in Power Systems. 13.1 Introduction. 13.2 Defi nition of Uncertainty. 13.3 Uncertainty Load Analysis. 13.4 Uncertainty Power Flow Analysis. 13.5 Economic Dispatch with Uncertainties. 13.6 Hydrothermal System Operation with Uncertainty. 13.7 Unit Commitment with Uncertainties. 13.8 VAR Optimization with Uncertain Reactive Load. 13.9 Probabilistic Optimal Power Flow. 13.10 Comparison of Deterministic and Probabilistic Methods. Author Biography. Index.

Mixed-integer nonlinear optimization

Abstract: Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.

On minimizing a convex function subject to linear inequalities

Abstract: SUMMARY THE minimization of a convex function of variables subject to linear inequalities is discussed briefly in general terms. Dantzig's Simplex Method is extended to yield finite algorithms for minimizing either a convex quadratic function or the sum of the t largest of a set of linear functions, and the solution of a generalization of the latter problem is indicated. In the last two sections a form of linear programming with random variables as coefficients is described, and shown to involve the minimization of a convex ftunction. Linear programming has been studied extensively in the last few years, as indicated by Vajda (1955). Various authors have mentioned the possibility of relaxing the requirement of linearity, but the practical problems of non-linear programming do not seem to have been considered in any detail.* This paper is concerned with some aspects of the simplest form of non-linear programming-the minimization of a convex function of variables subject to linear inequalities. In principle this can always be done using the method of steepest descents, but this will rarely be practical in its primitive form. We therefore consider special methods for some important particular classes of such functions. In Section 2 the Simplex Method, originally developed by Dantzig (1951) for linear programming, is outlined in terms sufficiently general to cover the applications to non-linear programming considered in the next two sections. In Section 3 we show how to minimize a convex quadratic function. This enables one to use what.amounts to the Newton-Raphson Method for minimizing a well-behaved general convex function: one finds a feasible solution of the constraints, and at each stage minimizes the quadratic function whose first and second derivatives at the feasible solution are the same as those of the given function.

Stochastic First- and Zeroth-Order Methods for Nonconvex Stochastic Programming

Abstract: In this paper, we introduce a new stochastic approximation type algorithm, namely, the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a postoptimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and we show that such modification allows us to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.