Nonconvex optimization of desirability functions

Desirability functions (DFs) are commonly used in optimization of design parameters with multiple quality characteristic to obtain a good compromise among predicted response models obtained from experimental designs. Besides discussing multi-objective approaches for optimization of DFs, we present a brief review of literature about most commonly used Derringer and Suich type of DFs and others as well as their capabilities and limitations. Optimization of DFs of Derringer and Suich is a challenging problem. Although they have an advantageous shape over other DFs, their nonsmooth nature is a drawback. Commercially available software products used by quality engineers usually do optimization of these functions by derivative free search methods on the design domain (such as Design-Expert), which involves the risk of not finding the global optimum in a reasonable time. Use of gradient-based methods (as in MINITAB) after smoothing nondifferentiable points is also proposed as well as different metaheuristics and interactive multi-objective approaches, which have their own drawbacks. In this study, by utilizing a reformulation on DFs, it is shown that the nonsmooth optimization problem becomes a nonconvex mixed-integer nonlinear problem. Then, a continuous relaxation of this problem can be solved with nonconvex and global optimization approaches supported by widely available software programs. We demonstrate our findings on two well-known examples from the quality engineering literature and their extensions.