A Heat-Transfer Optimization Problem

Abstract

<p> Discretization is an important tool to transfer optimization problems that include differentiations and integrals into standard optimization problems with a finite number of variables and a finite number of constraints. Recently, Betts and Campbell proposed a heat-transfer optimization problem that includes the heat partial differential equation as one of its constraints, and the objective function includes integrals of the temperature function squared. Using discretization methods, this problem can be converted to a convex quadratic optimization problem, which can be solved by standard interior point method solvers in polynomial time.</p> <p> The discretized model of the one dimensional problem is further analyzed, and some of its variants are studied. Extensive numerical testing is performed to demonstrate the power of the "discretize then optimize". Then the heat transfer optimization problem is generalized to two dimensions, and the discretized model and computational comparisons for this variant are included.</p> <p> Flexibility of discretization methods allow us to apply the same "diseretize then optimize" methodology to solve optimization problems that include differential and integral functions as constraints or objectives.</p>