Based on the stationary co-content theorem in non-linear circuit theory and the penalty function approach in non-linear programming theory, a canonical circuit for simulating general non-linear programming problems with equality and/or inequality constraints has been developed. The task of solving a non-linear optimization problem with constraints reduces to that of finding the solution of the associated canonical circuit using a circuit simulation program, such as SPICE.
A catalogue of canonical circuits is given for each class of non-linear programming problem. Using this catalogue, an engineer can solve non-linear optimization problems by a cook-book approach without learning any theory on non-linear programming. Several examples are given which demonstrate how SPICE can be used, without modification, for solving linear programming problems, quadratic programming problems, and polynomial programming problems.
1. Introduction
Non-linear programming is widely encountered in engineering problem^.'^ Indeed, almost any realistic design optimization problem subject to some practical constraints falls within the domain of non-linear programming. The essence of a general non-linear programming problem is to find the extremum of a non-linear objective function subject to certain constraints. From non-linear circuit theory we know that the operating point of a reciprocal circuit also corresponds to the extremum of some potential function under certain constraints (KCL, KVL and constitutive relations of circuit elements)'. Hence, if we can synthesize a reciprocal resistive non-linear circuit whose potential function is identical to the given objective function being minimized, and whose element constitutive relations impose the same equality and inequality constraints, then the solution of this circuit is precisely the solution of the non-linear programming problem. This observation was first given by Dennis6$ and subsequently extended by Stern.' However, their circuits contain circuit elements (e.g. d.c. multi-winding transformers and ideal diodes) which have been impractical to build until only recently. Moreover, their circuits are restricted to only two special classes of non-linear programming problems; namely, linear programming and quadratic programming.
The above objections can now be overcome as follows. First, since general purpose circuit simulation programs,' such as SPICE? are now widely available, the canonical 'non-linear programming' circuit can be simulated on a digital computer, thereby allowing the use of a much larger repertoire of circuit elements. Second, given any class of non-linear programming problem, not just linear or quadratic problems, we give a canonical circuit in this paper which simulates the problem.
The main objective of this paper is to present a catalogue of canonical non-linear programming circuits in a strictly cook-book fashion so that anyone having access to a circuit simulation program with a large enough repertoire of allowed circuit elements can solve a non-linear programming problem. Indeed, t Dennis (p. 1) gives the credit to S.