Approximating optimal solutions to biconvex parametric programs

Convex programming has been a research topic for a long time, both theoretically and algorithmically. Frequently, these programs lack complete data or contain rapidly shifting data. In response, we consider solving parametric programs, which allow for fast evaluation of the optimal solutions once the data is known. It has been established that, when the objective and constraint functions are convex in both variables and parameters, the optimal solutions can be estimated via linear interpolation. Many applications of parametric optimization violate the necessary convexity assumption. However, the linear interpolation is still useful; as such, we extend this interpolation to more general parametric programs in which the objective and constraint functions are biconvex. The resulting algorithm can be applied to scalarized multiobjective problems, which are inherently parametric, or be used in a gradient dual ascent method. We also provide two termination conditions and perform a numerical study on synthetic parametric biconvex optimization problems to compare their effectiveness.