Solving a Class of Nonlinear Programming Problems via a Homotopy Continuation Method

In this paper we present a homotopy continuation method for finding the Karush-Kuhn-Tucker point of a class of nonlinear non-convex programming problems. Two numerical examples are given to show that this method is effective. It should be pointed out that we extend the results of Lin et al. （see Appl. Math. Comput., 80（1996）, 209-224） to a broader class of non-convex programming problems.

HOMOTOPY CONTINUATION ALGORITHM FOR STRUCTURAL DAMAGE IDENTIFICATION

A new method is put forward for structural damage identification based on the homotopy continuation algorithm. A numerical example is presented to verify the method. The beams with different damage locations and different damage extents are identified by this method. The numerical examples have proved that this new method is capable of easy convergence, which is not sensitive to the initial iterative values. It is effective for accurately identifying multiple damages. By incorporating the finite element method into the homotopy continuation algorithm, the damage identifying ability of the new method can be greatly enhanced.

FindingSymbolicandAllNumericalSolutionsforDesignOptimizationBasedonMonotonicityAnalysisandSolvingPol

A method for determining symbolic and all numerical solutions in design optimization based on monotonicity analysis and solving polynomial systems is presented in this paper. Groebner Bases of the algebraic system equivalent to the subproblem of the design optimization is taken as the symbolic (analytical) expression of the optimum solution for the symbolic optimization, i.e. the problem with symbolic coefficients. A method based on substituting and eliminating for determining Groebner Bases is also proposed, and method for finding all numerical optimum solutions is discussed. Finally an example is given, demonstrating the strategy and efficiency of the method.

Numerical realization of the conditions of Max Nther's residual intersection theorem

The aim of this paper is to study numerical realization of the conditions of Max Nother＇s residual intersection theorem. The numerical realization relies on obtaining the inter- section of two algebraic curves by homotopy continuation method, computing the approximate places of an algebraic curve, getting the exact orders of a polynomial at the places, and determin- ing the multiplicity and character of a point of an algebraic curve. The numerical experiments show that our method is accurate, effective and robust without using multiprecision arithmetic, even if the coefficients of algebraic curves are inexact. We also conclude that the computational complexity of the numerical realization is polynomial time.

Computing Numerical Singular Points of Plane Algebraic Curves

Given an irreducible plane algebraic curve of degree d 〉 3, we compute its numerical singular points, determine their multiplicities, and count the number of distinct tangents at each to decide whether the singular points are ordinary. The numerical procedures rely on computing numerical solutions of polynomial systems by homotopy continuation method and a reliable method that calculates multiple roots of the univariate polynomials accurately using standard machine precision. It is completely different from the traditional symbolic computation and provides singular points and their related properties of some plane algebraic curves that the symbolic software Maple cannot work out. Without using multiprecision arithmetic, extensive numerical experiments show that our numerical procedures are accurate, efficient and robust, even if the coefficients of plane algebraic curves are inexact.

THE GLOBALIZATION OF DURAND-KERNER ALGORITHM

Making use of the theory of continuous homotopy and the relation betweensymmetric polynomtal and polynomtal in one variable the arthors devoted ims article to constructing a regularly homotopic curve with probability one. Discrete tracingalong this honlotopic curve leads 10 a class of Durand-Kerner algorithm with stepparameters. The convergernce of this class of algorithms is given, which solves theconjecture about the global property of Durand-Kerner algorithm. The.problem forsteplength selection is thoroughly discussed Finally, sufficient numerical examples areused to verify our theory

Automated Two-stage Conversion of Hourly Optimal DC Flow Solution to AC Power Flow

Conversion of hourly dispatch cases derived using DC optimal power flow(DCOPF)to AC power flow(ACPF)case is often challenging and requires arduous human analysis and intervention.This paper proposes an automated two-stage approach to solve ACPF formulated from DCOPF dispatch cases.The first stage involved the use of the conventional Newton Raphson method to solve the ACPF from flat start,then ACPF cases that are unsolvable in the first stage are subjected to a hotstarting incremental method,based on homotopy continuation,in the second stage.Critical tasks such as the addition of reactive power compensation and tuning of voltage setpoints that typically require human intervention were automated using a criteriabased selection method and optimal power flow respectively.Two datasets with hourly dispatches for the 243-bus reduced WECC system were used to test the proposed method.The algorithm was able to convert 100%of the first set of dispatch cases to solved ACPF cases.In the second dataset with suspect dispatch cases to represent an extreme conversion scenario,the algorithm created solved ACPF cases that satisfied a defined success criterion for 77.8%of the dispatch cases.The average run time for the hotstarting algorithm to create a solved ACPF case for a dispatch was less than 1 minute for the reduced WECC system.