General Nonconvex Research Articles

Optimal Power Flow by Improved Evolutionary Programming

This article proposes an improved evolutionary programming (IEP) for solving optimal power flow (OPF) with nonsmooth and nonconvex generator fuel cost curves. Initially, the whole population is divided into multiple subpopulations, which are used to perform the parallel search in divided solution space. IEP includes Gaussian and Cauchy mutation operators in different subpopulations to enhance the search diversity, selection operators with probabilistic updating strategy to avoid entrapping in local optimum, and reassignment operator for every subpopulation to exchange search information. The proposed IEP was tested on the IEEE 30 bus system with three different types of generator fuel cost curves. It is shown that IEP total generator fuel cost is less expensive than those of evolutionary programming, tabu search, hybrid tabu search and simulated annealing, and improved tabu search, leading to substantial generator fuel cost savings. Moreover, IEP can easily facilitate parallel implementation to reduce the computing time without sacrificing the quality of solution.

A Novel Tool for Multi-Model PID Controller Design

In this paper, it is shown how a robust perfomance Pill controller can be synthesized for a set of possible linear models. The method is carried out by solving a min-max optimization problem formulated in frequency domain. As the synthesis of a low-order controller for a high-order plant is in general nonconvex, the original optimization problem is divided into two convex optimization subproblems, which are solved iteratively until to achieve the complete convergence. These subproblems are constructed using a general two degree-of-freedom control configuration, in which all commercial industrial PID controllers can be converted.

An LMI approach to constrained optimization with homogeneous forms

This paper considers the problem of determining the minimum Euclidean distance of a point from a polynomial surface in R n . It is well known that this problem is in general non-convex. The main purpose of the paper is to investigate to what extent linear matrix inequality (LMI) techniques can be exploited for solving this problem. The first result of the paper shows that a lower bound to the global minimum can be achieved via the solution of a one-parameter family of linear matrix inequalities (LMIs). It is also pointed out that for some classes of problems the solution of a single LMI problem provides the lower bound. The second result concerns the tightness of the bound. It is shown that optimality of the lower bound amounts to solving a system of linear equations. An application example is finally presented to show the features of the approach.

Lagrange Multipliers for Nonconvex Generalized Gradients with Equality, Inequality, and Set Constraints

A Lagrange multiplier rule for finite dimensional Lipschitz problems that uses a nonconvex generalized gradient is proven. This result uses either both the linear generalized gradient and the generalized gradient of Mordukhovich or the linear generalized gradient and a qualification condition involving the pseudo-Lipschitz behavior of the feasible set under perturbations. The optimization problem includes equality constraints, inequality constraints, and a set constraint. This result extends known nonsmooth results for the Lipschitz case.

Graduated nonconvexity by functional focusing

Reconstruction of noise-corrupted surfaces may be stated as a (in general nonconvex) functional minimization problem. For functionals with quadratic data term, this paper addresses the criteria for such functionals to be convex, and the variational approach for minimization. I present two automatic and general methods of approximation with convex functionals based on Gaussian convolution. They are compared to the Blake-Zisserman graduated nonconvexity (GNC) method (1987) and Bilbro et al. (1992) and Geiger and Girosi's (1991) mean field annealing (MFA) of a weak membrane.

Turnpike Theorems in Nonconvex Nonstationary Environments

This paper provides a comprehensive development of turnpike theory in a stochastic aggregative model with time-varying nonconvex technology. A new approach to turnpike theorems is developed that exploits the monotonicity of optimal programs and utilizes a supermartingale process generated by stochastic Euler equations. This extends the classical turnpike theory to general nonconvex, nonstationary environments. Copyright 1997 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Nonconvex problems of global optimization: linear-quadratic control problems with quadratic constraints

In this paper, we study a class of optimization problems,which are of certain interest for control theory. These problemsare of global constrained optimization and may be nonconvex ingeneral. A simple approach to their solution is presented. Aspecial attention is paid to the case when the objective andconstraints functions are quadratic functionals on a Hilbertspace. As an example of an application of the general approach,a methodology is presented by which an optimal controller canbe synthesized for a finite-horizon linear-quadratic controlproblem with quadratic constraints. Both inequality and equalityconstraints are considered. The objective and constraints functionalsmay be nonconvex and may contain both integral and terminal summands.It is shown that, under certain assumptions, the optimal controlexists, is unique, and has feedback structure. Furthermore, theoptimal controller can be computed by the methods of classiclinear-quadratic control theory coupled with those of finitedimensional convex programming.

A Deterministic Global Design Optimization Method for Nonconvex Generalized Polynomial Problems

A new design optimization method is described for finding global solutions of models with a nonconvex objective function and nonlinear constraints. All functions are assumed to be generalized polynomials. By introducing new variables, the original model is transformed into one with a linear objective function, one convex and one reversed convex constraint. A two-phase algorithm that includes global feasible search and local optimal search is used for globally optimizing the transformed model. Several examples illustrate the method.

Large Scale Dynamic Programming Based Dispatch Including Transmission Losses

A new technique for economic dispatch based on the principle of dynamic programming and including both transmission limits and loss representation is presented. Despite the general view that DP is inherently time consumming and requires enormous computer memory, the method described here has neither of these disadvantages. It has many desirable characteristics including numerical stability, high speed, high accuracy and superior capability in handling non-linear, non-convex generation costs. Individual line flow constraints are considered and the costs of transmission losses are included within the overall minimization stategy. Generating units in a station connected to the same busbar need not neccessarily be lumped into a single entity but can be modelled separately. Because of its speed, the approach is potentially suitable for on-line application even for a very large system. The theoretical derivation of the method is presented and numerical results on applications to various networks including a data set from CEGB are reported.

Generalised cut map algorithms for tolerance and tolerance-tuning problems

The problem considered is the choice of a set of system parameters, so that inequality constraints are satisfied or can be satisfied by tuning, for a specified variation of parameter values about their nominal value. Such problems occur when systems must be synthesised from components whose values are known only to certain tolerances. Fully implementable algorithms that are suitable for the general nonconvex cases and utilise concepts employed by Eaves and Zangwill in their generalised cutting plane algorithms are presented. The algorithms generalise earlier algorithms for the pure tolerance (no tuning) and the tolerance-tuning problems that are suitable only for the case when the tolerance and tuning regions are constant.

MEAN-FIELD THEORY AND COMPUTATION OF ELECTROSTATICS WITH IONIC CONCENTRATION DEPENDENT DIELECTRICS.

We construct a mean-field variational model to study how the dependence of dielectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and the ionic ideal-gas entropy. The electrostatic potential is determined by Poisson's equation in which the dielectric coefficient depends on the sum of concentrations of individual ionic species. This dependence is assumed to be qualitatively the same as that on the salt concentration for which experimental data are available and analytical forms can be obtained by the data fitting. We derive the first and second variations of the free-energy functional, obtain the generalized Boltzmann distributions, and show that the free-energy functional is in general nonconvex. To validate our mathematical analysis, we numerically minimize our electrostatic free-energy functional for a radially symmetric charged system. Our extensive computations reveal several features that are significantly different from a system modeled with a dielectric coefficient independent of ionic concentration. These include the non-monotonicity of ionic concentrations, the ionic depletion near a charged surface that has been previously predicted by a one-dimensional model, and the enhancement of such depletion due to the increase of surface charges or bulk ionic concentrations.