Solution Sets of Convex Programs Related to Chemical Equilibrium Problems

We consider the program (I) infF(x), subject to x ∈ H and Ax = b, where H is any convex set in the positive orthant, A is any m × n matrix, b is any m-vector, and F is convex and continuous on H. A subsidiary program is used to study the nature of the constraint set and optimal solution set of program (I). The method used involves classifying duality states of the program with respect to duality states of its subsidiary program. Extensions are thereby obtained of Bigelow-DeHaven-Shapiro results [SIAM J. Appl. Math. 18, 768–775 (1970)] on the set of solutions to chemical equilibrium problems. An example of Bigelow and Shapiro is given illustrating the states by means of a simple, ideal, one-phase chemical-equilibrium problem in two dimensions.