## Sequential gradient-restoration algorithm for mathematical programming problems with inequality constraints

##### Author

Sims, Edward Michael

##### Date

1983##### Advisor

Miele, Angelo

##### Degree

Master of Science

##### Abstract

The problem of minimizing a function f(x) of an n-vector x, subject to q equality constraints <{>(x) = and ninequality constraints w(x) _> , is considered. An algorithm of the sequential gradient-restoration type is developed. It involves the alternate succession of gradient phases and restoration phases. For general inequalities, each iteration of the gradient phase and the restoration phase requires the solution of a linear system of order q + n, where q denotes the number of equality constraints and n the number of inequality constraints. The unknowns of the linear systems are the q components of the multiplier X associated with the equality constraints and the n components of the multiplier p associated with the inequality constraints. Considerable simplifications are possible if the inequalities w(x) >_ have a special form. In this connection, the following cases are studied: (PI) lower bounds on x; and (P2) upper and lower bounds on x. If one exploits the special structure of Problems (PI) and (P2) and the properties of diagonal matrices, the algorithmic work per iteration can be reduced considerably. This is due to the fact that the multipliers X and p need not be computed simultaneously, but can be computed sequentially. Specifically, the multiplier X is determined by a linear equation of order q; then, the multiplier pis determined through subsequent multiplications. This means that, for Problems (PI) and (P2), the algorithmic work per iteration is about the same as the algorithmic work per iteration occurring in Problem (P): minimize a function f(x) of an n-vector x, subject to equality constraints <f>(x) = .