Methods of Feasible Directions: A Review

Since the theoretical basis for the method of feasible directions (MFD) was originally developed by Zoutendijk in 1960’s, several basic variations and modifications of MFD were proposed and investigated. Even though faster algorithms for solving nonlinear programming problems exist, MFD has never been abandoned because of several important advantages such as the descent property, feasibility of all iterates, conceptual simplicity and overall computational efficiency. In fact, MFD is especially popular in the engineering community where it is heavily used in structural optimization and other engineering design optimization. In engineering design, it is extremely important to end up with a design which satisfies the hard specifications expressed by a set of inequalities. This paper reviews several typical methods of feasible directions and their variants: Zoutendijk’s MFD, Topkis — Veinott’s MFD, Pironneau — Polak’s MFD, Wiest — Polak’s MFD, Cawood — Kostreva’s MFD, Panier — Tits’ FSQP and a modified MFD. Then, the performances of three MFD: Pironneau — Polak’s MFD, Cawood — Kostreva’s MFD and the modified MFD are compared numerically and graphically. The results suggest that the modified MFD is more efficient than the other two MFD.

[1]  Michael M. Kostreva,et al.  A generalization of the norm-relaxed method of feasible directions , 1999, Appl. Math. Comput..

[2]  Elijah Polak,et al.  Computational methods in optimization , 1971 .

[3]  David Q. Mayne,et al.  Combined phase I—phase II methods of feasible directions , 1979, Math. Program..

[4]  Philip E. Gill,et al.  The computation of Lagrange-multiplier estimates for constrained minimization , 1979, Math. Program..

[5]  José Herskovits,et al.  A two-stage feasible directions algorithm for nonlinear constrained optimization , 1981, Math. Program..

[6]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[7]  M. M. Kostreva,et al.  Self-Tuning Norm-Relaxed Method of Feasible Directions , 1997 .

[8]  G. Zoutendijk,et al.  Methods of Feasible Directions , 1962, The Mathematical Gazette.

[9]  R. W. Chaney.  On the Pironneau-Polak method of centers , 1976 .

[10]  André L. Tits,et al.  On combining feasibility, descent and superlinear convergence in inequality constrained optimization , 1993, Math. Program..

[11]  M. Kostreva,et al.  Norm-relaxed method of feasible directions for solving nonlinear programming problems , 1994 .

[12]  M. J. D. Powell,et al.  A fast algorithm for nonlinearly constrained optimization calculations , 1978 .

[13]  E. Panier,et al.  A QP-Free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization , 1988 .

[14]  G. Vanderplaats.  An efficient feasible directions algorithm for design synthesis , 1984 .

[15]  J. Korycki,et al.  Norm-relaxed method of feasible directions: application in structural optimization , 1996 .

[16]  M. M. Kostreva,et al.  Convergence analysis of norm-relaxed method of feasible directions , 1996 .

[17]  Elijah Polak,et al.  On the rate of convergence of certain methods of centers , 1972, Math. Program..

[18]  A. F. Veinott,et al.  On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming , 1967 .

[19]  E. Polak,et al.  Rate of Convergence of a Class of Methods of Feasible Directions , 1973 .

[20]  Surya N. Patnaik,et al.  An optimization algorithm based on the method of feasible directions , 1995 .

[21]  R. W. Chaney.  On the rate of convergence of some feasible direction algorithms , 1976 .

[22]  E. Panier,et al.  A superlinearly convergent feasible method for the solution of inequality constrained optimization problems , 1987 .

[23]  E. Polak,et al.  A generalized quadratic programming-based phase I-phase II method for inequality-constrained optimization , 1992 .

[24]  A. Tits,et al.  Enhanced methods of feasible directions for engineering design problems , 1986 .

[25]  Jack A. Korycki,et al.  Norm-Relaxed Method of Feasible Directions , 1995 .

[26]  Garret N. Vanderplaats,et al.  Numerical optimization techniques for engineering design , 1999 .