Abstract
In this note partially separable convex programs are dualized in such a way that, under certain assumptions, unconstrained concave duals arise. A return formula is given by which the solution of the primal is directly computed if a solution of the dual is known. Further, the solvability of both the primal and the dual is shown to depend essentially on the behaviour of the lower dimensional programs for determining the Fenchel conjugates.
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References
W. Burmeister, W. Heß and J.W. Schmidt, “Convex spline interpolants with minimal curvature,”Computing 35 (1985) 219–229.
S. Dietze and J.W. Schmidt, “Determination of shape preserving spline interpolants with minimal curvature via dual programs,”Journal of Approximation Theory 52 (1988) 43–57.
S. Dietze and J.W. Schmidt, “Unconstrained dual programs for partially separable constrained optimization problems,” Preprint 07-10-89, Technical University of Dresden (Dresden, Germany, 1989).
H. Everett, “Generalized Lagrange multiplier method for solving problems of optimum allocation of resources,”Operations Research 11 (1963) 399–417.
J.E. Falk, “Lagrange multipliers and nonlinear programming,”Journal of Mathematical Analysis and Applications 19 (1967) 141–159.
A. Griewank and Ph.L. Toint, “On the unconstrained optimization of partially separable functions,” in: M.J.D. Powell, ed.,Nonlinear Optimization 1981 (Academic Press, New York, 1982) pp. 301–312.
A. Griewank and Ph.L. Toint, “On the existence of convex decompositions of partially separable functions,”Mathematical Programming 28 (1984) 25–49.
M. Krätzschmar, “A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems,”Numerische Mathematik 54 (1989) 507–531.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
J.W. Schmidt, “An unconstrained dual program for computing convexC 1-spline approximants,”Computing 39 (1987) 133–140.
J.W. Schmidt, “On tridiagonal linear complementarity problems,”Numerische Mathematik 51 (1987) 11–21.
J.W. Schmidt, “Specially structured convex optimization problems: Computational aspects and applications,” in: D. Greenspan, P. Rózsa, eds.,Numerical Methods, Colloquia Mathematica Societatis János Bolyai, Vol. 50 (North-Holland, Amsterdam, 1987) pp. 565–579.
J.W. Schmidt, “Dual algorithms for solving convex partially separable optimization problems,”Jahresberichte des Deutschen Mathematiker-Vereinigung 94 (1992).
J.W. Schmidt and W. Heß, “Positivity of cubic polynomials on intervals and positive spline interpolation,”BIT (Nordisk Tidskrift for Informationsbehandling) 28 (1988) 340–352.
J.W. Schmidt and I. Scholz, “A dual algorithm for convex-concave data smoothing by cubicC 2-splines,”Numerische Mathematik 57 (1990) 333–350.
S. Ulm, “On two-level optimization,”Eesti NSV Teaduste Akadeemia Toimetised Füüsika-Matemaatika 18 (1969) 3–13.
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Schmidt, J.W., Dietze, S. Unconstrained duals to partially separable constrained programs. Mathematical Programming 56, 337–341 (1992). https://doi.org/10.1007/BF01580906
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DOI: https://doi.org/10.1007/BF01580906