Abstract
We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.
Similar content being viewed by others
References
A. Bachem and B. Korte, “Mathematical programming and the estimation of input—output matrices,” in: G. Gastellani and P. Mazzolani, ed.,Mathematical Programming and Its Economic Applications (Angeli, Milano, 1981).
M.S. Bazara and C.M. Shetty,Nonlinear Programming, Theory and Algorithms (John Wiley & Sons, NY, 1979).
D.P. Bertsekas, “On the Golstein—Levtin—Poljak gradient projection method,”IEEE Transactions on Automatic Control 21 (1976) 174–184.
R. Burkard, H.W. Hamacher and G. Rote, “Sandwich approximation of univariate convex functions with an application to separable convex programming,”Naval Research Logistics 38 (1991) 911–924.
A. Charnes and C.E. Lemke, “Minimization of nonlinear separable convex functionals,”Naval Research Logistics Quarterly 1 (1954) 301–312.
M.G. Cox, “An algorithm for approximating convex functions by means of first degree splines,”The Computer Journal 14 (1971) 272–275.
R. Fourer, “A simplex algorithm for piecewise-linear programming I: Derivation and proof,”Mathematical Programming 33 (1985) 204–233.
R. Fourer, “A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy,”Mathematical Programming 41 (3) (1988) 281–315.
R. Fourer, “A simplex algorithm for piecewise-linear programming III: Computational analysis and applications,”Mathematical Programming 53 (1992) 213–235.
R. Fourer and R.E. Marsten, “Solving piecewise-linear programs: Experiment with a simplex approach,”ORSA Journal on Computing 4 (1992) 16–31.
A.M. Geoffrion, “Objective function approximations in mathematical programming,”Mathematical Programming 13 (1977) 23–37.
F. Güder, “Solution of multi-commodity intertemporal spatial equilibrium problems,” Ph.D. Dissertation, University of Wisconsin-Madison, 1983.
F. Güder and J.G. Morris, “Objective function approximation: An application to spatial price equilibrium models,”American Journal of Agricultural Economics 70 (1988) 391–396.
G. Hadley,Nonlinear and Dynamic Programming (Addison-Wesley, Reading, MA, 1964).
C.Y. Kao and R.R. Meyer, “Secant approximation methods for convex optimization,”Mathematical Programming Study 14 (1981) 143–162.
R.F. Love, “The optimal design of a container intoN compartments by convex programming,”The International Journal of Production Research 7 (1969) 199–206.
R.R. Meyer, “Two-segment separable programming,”Management Science 25 (1979) 385–395.
R.R. Meyer, “Computational aspects of two-segment separable programming,”Mathematical Programming 26 (1983) 21–39.
G.M. Phillips, “Algorithms for piecewise straightline approximations,”The Computer Journal 11 (1968) 211–212.
R.T. Rockafellar,Network Flows and Monotropic Optimization (John Wiley and Sons Inc., New York, 1984).
J. Sun and K. Tsai, “An implementation of the network simplex method for piecewise-linear programs,” Technical Report 89-07, Dept. of Industrial Engineering and Management Sciences, Northwestern Univ., Evanston, IL 60208, 1989.
L.S. Thakur, “Error analysis for convex separable programs: The piecewise linear approximation and the bounds on the optimal objective value,”SIAM Journal on Applied Mathematics 34 (1978) 704–714.
L.S. Thakur, “Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions,”Journal of Mathematical Analysis and Applications 75 (1980) 486–494.
L.S. Thakur, “Solving highly nonlinear convex separable programs using successive approximation,”Computers and Operations Research Vol II, 2 (1984) 113–128.
L.S. Thakur, “Successive approximation in separable programming: An improved procedure for convex separable programs,”Naval Research Logistics 33 (1986) 325–358.
Author information
Authors and Affiliations
Additional information
Corresponding author.
Rights and permissions
About this article
Cite this article
Güder, F., Morris, J.G. Optimal objective function approximation for separable convex quadratic programming. Mathematical Programming 67, 133–142 (1994). https://doi.org/10.1007/BF01582218
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01582218