Using Expression Graphs in Optimization Algorithms

Abstract

An expression graph, informally speaking, represents a function in a way that can be manipulated to reveal various kinds of information about the function, such as its value or partial derivatives at specified arguments and bounds thereon in specified regions. (Various representations are possible, and all are equivalent in complexity, in that one can be converted to another in time linear in the expression’s size.) For mathematical programming problems, including the mixed-integer nonlinear programming problems that were the subject of the IMA workshop that led to this paper, there are various advantages to representing problems as collections of expression graphs. “Presolve” deductions can simplify the problem, e.g., by reducing the domains of some variables and proving that some inequality constraints are never or always active. To find global solutions, it is helpful sometimes to solve relaxed problems (e.g., allowing some “integer” variables to vary continuously or introducing convex or concave relaxations of some constraints or objectives), and to introduce “cuts” that exclude some relaxed variable values. There are various ways to compute bounds on an expression within a specified region or to compute relaxed expressions from expression graphs. This paper sketches some of them. As new information becomes available in the course of a branch-and-bound (or -cut) algorithm, some expression-graph manipulations and presolve deductions can be revisited and tightened, so keeping expression graphs around during the solution process can be helpful. Algebraic problem representations are a convenient source of expression graphs. One of my reasons for interest in the AMPL modeling language is that it delivers expression graphs to solvers.

AMS(MOS) subject classifications. Primary 68U01, 68N20, 68W30, 05C85.