By Berc Rustem
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Extra info for Projection Methods in Constrained Optimisation and Applications to Optimal Policy Decisions
Example text
2). 1 The Feasible Region Given an infeasible i n i t i a l point 2~u, the problem is to determine a feasible point x ( R where the region R is defined as R A {X E Em I NT X ~ = -- mo-- } . I) o The subscript mo denotes the total number of constraints. At an infeasible point Xu~ En some of the constraints describing R may be violated, others just satisfied as equalities and the rest satisfied as s t r i c t inequalities. The set of the indices of a l l the constraints violated at ~u w i l l be defined as V(Xu) a {v I
16 inverse Hessian of the objective function in a given subspace ~ c En. 3 1 Methods Based on Computing Bases for In this seotion methods using approximations of Gkor HK = GKI or operators that involve Hk w i l ] be discussed. These methods are based on solutions to problem ( l . l . S l ) in Section ( l . l . 2 ) and thus use ( I . I . 5 0 ) to compute ~l" The f i r s t algorithm that applied projections to constrained optimisation was Rosen's Gradient Projection method (Rosen (1960)). The descent directions and the Lagrange multipliers generated by this algorithm may be obtained by setting Gk = I in ( l .
I . 4 5 ) may therefore be set equal to the vectors ~ l , i = m+l, . . , n, i . e . 25 z: [ _tm+iD . . 74) . 60) "m~q = Nm:V ~ = TT : vq(_d;) = G_d; + vf(~k) . 75) To invert TT the identity (TT)-I = [Nm k I, = [ ..... Z]"..... is used. 76) zT is obtained. 76). 49). The f i r s t method to use T-I to generate the matrix Z was the reduced-gradient method of Wolfe (Ig6/). In this method the columns of VT are selected from the normals of inactive constraints. 49) even when Gk = I . An alternative choice for V is provided by the variable reduction method due to McCormick (1970Da).