Symbolic design optimization : a computer aided method to increase monotonicity through variable reformulation

Monotonicity analysis, developed by [Wilde 78] and [Papalambros 79], is an approach to simplifying and solving some nonlinear, constrained global optimization problems without iterative numerical calculations. Unfortunately, monotonicity analysis is limited to problems in which the objective function and constraints vary roonotonically with the design variables. To alleviate this limitation it is sometimes possible to reformulate the design problem to increase the degree of monotonicity and thereby facilitate the complete or partial application of monotonicity analysis procedures. These useful reformulations are accomplished by a transformation to alternative design parameters, such as a critical ratio, a nondimensional parameter, or a simple difference; e.g. the ratio of surface area to volume for heat transfer loss, the Reynold's number in fluid mechanics, or the velocity difference across a fluid coupling. We have developed a method by which the alternative parameters are chosen for physical significance and for the ability to reduce the number of nonmonotonic variables in a system of constraints. Rules have been developed for the creation of physically significant new parameters from the algebraic combination of the original parameters. The rules are based on engineering principles and rely on knowledge about what a parameter physically represents rather than other qualities such as dimensions. A computer based system, called EUDOXUS, has been developed to automate this procedure. The method and its implementation have demonstrated successful results for highly nonlinear, nonmonotonic, and coupled parameterized designs in various mechanical engineering domains. Introduction In this paper we ait concerned with parametric design problems that can be expressed in tenns of n design variables, V = [uv w^ ...» um]. The optimization problem is to minimize or maximize an objective function: /o(0> (1) subject to k inequality constraints and r equality constraints: f^U) <> 0 i = l , * (2) fj(U) = 0 y = *+l,*+r (3) In most cases the number of unknowns, n, is greater than the number of equality constraints, r, so that the problem is underconstrained. Optimal, rather than merely satisfactory solutions to design problems are often identified intuitively by experienced and insightful engineers. This is true even when the design constraints are very complex because optimal designs are frequently determined by the inequality constraints which delimit the feasible design space. In these cases, finding an optimal solution requires the identification of the inequality constraints which are satisfied as equalities at the optimum. This is often possible when the objective and the constraints vary monotonically with respea to some or all of the unknowns. A procedure known as monotonicity analysis facilitates the identification of the critical constraints at the optimum. The necessary conditions for the optimality of monotonic systems, first explicated by Wilde and Papalambros [Wilde 78, Wilde 86, Papalambros 79, Papalambros 88], can be summarized in the following two rules paraphrased from [Agogino 87]: Rule 1: When an objective function is monotonic wkh respect to a variable parameter then there exists at least one active constraint at the optimum which bounds the variable in a direction opposite of the objective. Rule 2: When a variable is not contained in the objective function then it must either be bounded from both above and below by active constraints at the optimum or not bounded at all such that all constraints which are monotonic with respect to that variable are inactive and can be removed from the problem definitioa These rules can be used to solve some optimization problems symbolically (and automatically [Agogino 87]). The active constraints which limit the degree to which the objective function can be minimized or maximized become explicit. Some of the drawbacks of numerical techniques, e.g. convergence and local optima, are avoided and more importantly valuable qualitative information is obtained. Unfortunately, monotonicity analysis is limited to design optimization problems where the objective function and constraints are monotonic either globally or over a predefined regional domain. In the next section we illustrate, using a simple truss design problem, how reformulations of design problems involving a transformation of variable can increase the number and degree of monotonicity assignments thereby improving symbolic optimization procedures. Although an unlimited number of reformulations are possible, those which are most useful exhibit three characteristics: 1. a reduction in the number of variables in some of the constraints and the objective function, 2. an increase in the degree of constraint and objective monotonicity, 3. and are based on new variables which are physically meaningful. New variables with physical significance are especially important for establishing qualitative insights and for making meaningful estimates when numerical methods are used. Following the details of the example we concentrate on the methods used to identify reformulations as driven by these attributes. In short, we take a two step approach. First, we discuss a method to create physically meaningful new parameters based on the physical meaning of the original parameters. The candidate parameters must then be grouped into transformation sets that produce a more monotonic foimulatioa The second step is to assemble the new parameters into basis sets, perform transformations of variable, and evaluate the utility of the resulting problem reformulations. Although these techniques are independent of any computer implementation they are, in general, computationally intensive. These ideas are implemented in a computer program called EUDOXUS. The final part of this paper discusses the program implementation and effectiveness. Figure 1: Two Bar Truss Configuration 'Eudoxus of Cnidus (b. 408 BCt d. 355 BC) was a Greek scholar who made contributions in mathematics, astronomy, geography, philosophy, and law. His theory of proportions solved the crisis of the Pythagorean discovery of irrational numbers and his method of exhaustion was a forerunner to modern calculus. Two Bar Truss Example A stnictuie must be designed to suppoit horizontal and veitical loads at a minimum height. A simple truss of the type shown in Figure 1 is to be considered [Fox 71, Parkinson 85]. The truss consists of a pair of tubes pinned together and to ground supports. The structure must withstand a veitical load, Pv, and a horizontal load, Ph% without failing by yielding or buckling, and without excessive veitical deflection, 5^^. Additional constraints enforce the requirement on minimum height, Hmin$ and maintain an outside diameter greater than the inside diameter. The parameters describing the truss are the halfspan, B, height, //, tube outside tube diameter, do, and the inside diameter, dv The material properties, modulus of elasticity, £, yield stress, oyig[d9 and density, p, are known. Mathematically, the problem is to minimize truss mass: