Complementary Pivot Theory of Mathematical Programming

1 The fundamental problem can be extended from p sets each consisting of a pair of variables only one of which can bc nonbasic to k sets of several variables each, only one of which can be nonbasic. To be specific, consider a system w = q f Nz, zw > 0, z > 0, where N is a p x k matrix (k < p) and the variables wl, , zu@ are partitioned into k nonempty sets SI, I = I, ., k. Let TI = SJ IJ {q}, I = 1, , k. We seek a solution of the system in which exactly one member of each set T, is nonbasic. (The fundamental problem is of this form where k = p and Tl = {q, q}.) The underlying idea of Lemke’s approach (Sectlon 2) applies here. For example, it can be shown that this problem has a solution when N > 0. A paper is currently being prepared for publication in which this extension is developed in detail. * In general, capital italic letters denote matrices while vectors are denoted by lower case italic letters. Whether a vector is a row or a column will always be clear from the context, and consequently we dispense with transpose signs on vectors. In (2), for example, zw represents the scalar product of z (row) and w (column). The superscript T indicates the transpose of the matrix to which it is affixed.