Path-following methods for Kuhn-Tucker curves by an active index set strategy

Abstract

In this paper Kuhn-Tucker curves of one-parameter optimization problems P(t), t ~ ~ are discussed. Using an active index set strategy, path following procedures for P(t) are developed. One-parameter optimization problems naturally arise if a nonlinear optimization problem P is solved by a homotopy method. Under certain conditions on P it is shown that the proposed homotopy method is constructive with probability one in the sense of Chow, Mallet-Parer, Yorke [4].

Continuation and path following

“…The local sensitivity of such systems has been analysed, e.g., in Fiacco (1983Fiacco ( , 1984 and Robinson (1987). Many authors have used bifurcation and singularity theory to investigate the local behaviour and persistence of minima at the singular points of this system, see, e.g., Bank, Guddat, Klatte, Kummer and Tammer (1983), Gfrerer, Guddat and Wacker (1983), Gfrerer, Guddat, Wacker and Zulehner (1985), Guddat, Guerra Vasquez and Jongen (1990), Guddat, Jongen, Kummer and Nozicka (1987), Twilt (1983, 1986), Jongen and Weber (1990), Kojima and Hirabayashi (1984) and Tiahrt (1987, 1990). Rakowska, Haftka and Watson (1991) discuss algorithms for tracking paths of optimal solutions.…”

Pathfollowing methods in nonlinear optimization III: Lagrange multiplier embedding

“…e.g. [5,8,14]). Consider a one-parametric optimization problem /3(t) min{f(y, t)ly e 34(t)}, t ~ [0, 1] , (1.1) where…”

On the calculation of a feasible point of a nonconvex set: pathfollowing with jumps

“…The concept of pathfollowing with jumps was introduced and applied in [1][2][3][4][5][6]. There exists a variety of applications and numerical results for pathfollowing (or homotopy or continuation) methods; we refer, e.g., to [7][8][9][10][11][12][13][14][15][16].…”