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.

[1]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[2]  Anthony V. Fiacco,et al.  Generalized convexity and concavity of the optimal value function in nonlinear programming , 1987, Math. Program..

[3]  Lakshman S. Thakur,et al.  Error Analysis for Convex Separable Programs: The Piecewise Linear Approximation and The Bounds on The Optimal Objective Value , 1978 .

[4]  Abolfazl Ghaemi,et al.  Computable stability analysis techniques for nonlinear programming: sensitivities, optimal value bounds, and applications , 1981 .

[5]  Stephen M. Robinson,et al.  Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear-programming algorithms , 1974, Math. Program..

[6]  Arthur M. Geoffrion,et al.  Objective function approximations in mathematical programming , 1977, Math. Program..

[7]  J. Kyparisis,et al.  Sensitivity analysis in posynomial geometric programming , 1988 .

[8]  Donald M. Topkis,et al.  Rates of Stability in Nonlinear Programming , 1976, Oper. Res..

[9]  R. Meyer Two-Segment Separable Programming , 1979 .

[10]  Anthony V. Fiacco,et al.  A User's Manual for SENSUMT: A Penalty Function Computer Program for Solution, Sensitivity Analysis, and Optimal Value Bound Calculation in Parametric Nonlinear Programs. , 1980 .

[11]  Gary A. Kochenberger,et al.  Sensitivity Analysis Procedures for Geometric Programs: Computational Aspects , 1978, TOMS.

[12]  Garth P. McCormick,et al.  Locating an Isolated Global Minimizer of a Constrained Nonconvex Program , 1980, Math. Oper. Res..

[13]  James W. Daniel,et al.  Stability of the solution of definite quadratic programs , 1973, Math. Program..

[14]  H. P. Benson,et al.  Algorithms for parametric nonconvex programming , 1982 .

[15]  Lakshman S. Thakur,et al.  Solving highly nonlinear convex separable programs using successive approximation , 1984, Comput. Oper. Res..

[16]  Garth P. McCormick,et al.  Bounding Global Minima , 1976, Math. Oper. Res..

[17]  A. Fiacco,et al.  Convexity and concavity properties of the optimal value function in parametric nonlinear programming , 1983 .

[18]  S. M. Robinson Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[19]  Stephen M. Robinson,et al.  Computable error bounds for nonlinear programming , 1973, Math. Program..

[20]  Anthony V. Fiacco,et al.  Sensitivity Analysis of a Nonlinear Water Pollution Control Model Using an Upper Hudson River Data Base , 1982, Oper. Res..

[21]  Anthony Fiacco Computable Optimal Value Bounds and Solution Vector Estimates for General Parametric NLP Programs. , 1981 .

[22]  Garth P. McCormick,et al.  Bounding Global Minima with Interval Arithmetic , 1979, Oper. Res..

[23]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[24]  Anthony V. Fiacco,et al.  Sensitivity analysis of a nonlinear structural design problem , 1982, Comput. Oper. Res..

[25]  S. M. Robinson Generalized equations and their solutions, Part I: Basic theory , 1979 .

[26]  J. B. Rosen,et al.  Inequalities for Stochastic Nonlinear Programming Problems , 1964 .

[27]  S. M. Robinson Bounds for error in the solution set of a perturbed linear program , 1973 .