Global multi-parametric optimal value bounds and solution estimates for separable parametric programs

In this tutorial, a strategy is described for calculating parametric piecewise-linear optimal value bounds for nonconvex separable programs containing several parameters restricted to a specified convex set. The methodology is based on first fixing the value of the parameters, then constructing sequences of underestimating and overestimating convex programs whose optimal values respectively increase or decrease to the global optimal value of the original problem. Existing procedures are used for calculating parametric lower bounds on the optimal value of each underestimating problem and parametric upper bounds on the optimal value of each overestimating problem in the sequence, over the given set of parameters. Appropriate updating of the bounds leads to a nondecreasing sequence of lower bounds and a nonincreasing sequence of upper bounds, on the optimal value of the original problem, continuing until the bounds satisfy a specified tolerance at the value of the parameter that was fixed at the outset. If the bounds are also sufficiently tight over the entire set of parameters, according to criteria specified by the user, then the calculation is complete. Otherwise, another parameter value is selected and the procedure is repeated, until the specified criteria are satisfied over the entire set of parameters. A parametric piecewise-linear solution vector approximation is also obtained. Results are expected in the theory, computations, and practical applications. The general idea of developing results for general problems that are limits of results that hold for a sequence of well-behaved (e.g., convex) problems should be quite fruitful.