Abstract
In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.
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References
A. Auslender, “Minimisation de fonctions localement lipschitziennes: Applications a la programmation mi-convexe, mi-differentiable,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming, Vol. 3 (Academic Press, New York, 1978) pp. 429–460.
A. Auslender, “Numerical methods for nondifferentiable convex optimization,”Mathematical Programming Study 30 (1987) 102–126.
D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).
F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
L. Cromme, “Strong uniqueness: A far-reaching criterion for the convergence analysis of iterative procedures,”Numerische Mathematik 29 (1978) 179–193.
R.S. Dembo and T. Steihaug, “Truncated-Newton algorithms for large-scale unconstrained optimization,”Mathematical Programming 26 (1983) 190–212.
R. Fletcher,Practical Methods of Optimization, Vol. 2: Constrained Optimization (Wiley, Chichester, 1981).
M. Fukushima, “A descent algorithm for nonsmooth convex optimization,”Mathematical Programming 30 (1984) 163–175.
M. Fukushima and H. Mine, “A generalized proximal point algorithm for certain nonconvex minimization problems,”International Journal of Systems Science 12 (1981) 989–1000.
J.-B. Hiriart-Urruty, “ε-Subdifferential calculus,” in: J.-P. Aubin and R.B. Vinter, eds.,Convex Analysis and Optimization, Research Notes in Mathematics Series, Vol. 57 (Pitman, London, 1982) pp. 43–92.
J.-B. Hiriart-Urruty, “Calculus rules on the approximate second-order directional derivative of a convex function,”SIAM Journal on Control and Optimization 22 (1984) 381–404.
K.C. Kiwiel, “An exact penalty function algorithm for nonsmooth convex constrained minimization problems,”IMA Journal of Numerical Analysis 5 (1985) 111–119.
K.C. Kiwiel, “A method for minimizing the sum of a convex function and a continuously differentiable function,”Journal of Optimization Theory and Applications 48 (1986) 437–449.
K.C. Kiwiel, “A constraint linearization method for nondifferentiable convex minimization,”Numerische Mathematik 51 (1987) 395–414.
C. Lemarechal and J. Zowe, “Some remarks on the construction of higher order algorithms in convex optimization,”Applied Mathematics and Optimization 10 (1983) 51–68.
H. Mine and M. Fukushima, “A minimization method for the sum of a convex function and a continuously differentiable function,”Journal of Optimization Theory and Applications 33 (1981) 9–23.
E. Polak, D.Q. Mayne and Y. Wardi, “On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems,”SIAM Journal on Control and Optimization 21 (1983) 179–203.
M.J.D. Powell, “Variable metric methods for constrained optimization,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming: The State of the Art, Bonn 1982 (Springer, Berlin-Heidelberg, 1983) pp. 288–311.
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar, “Monotone operators and the proximal point algorithm,”SIAM Journal on Control and Optimization 14 (1976) 877–898.
J.F. Shapiro,Mathematical Programming: Structures and Algorithms (Wiley, New York, 1979).
J.E. Spingarn, “Submonotone mappings and the proximal point algorithm,”Numerical Functional Analysis and Optimization 4 (1981/1982) 123–150.
J.-J. Strodiot, V.H. Nguyen and N. Heukemes, “ε-Optimal solutions in nondifferentiable convex programming and some related questions,”Mathematical Programming 25 (1983) 307–328.
R.S. Womersley, “Optimality conditions for piecewise smooth functions,”Mathematical Programming Study 17 (1982) 13–27.
R.S. Womersley, “Local properties of algorithms for minimizing nonsmooth composite functions,”Mathematical Programming 32 (1985) 69–89.
R.S. Womersley and R. Fletcher, “An algorithm for composite nonsmooth optimization problems,”Journal of Optimization Theory and Applications 48 (1986) 493–523.
E. Yamakawa, M. Fukushima and T. Ibaraki, “An efficient trust region algorithm for minimizing nondifferentiable composite functions,”SIAM Journal on Scientific and Statistical Computing 10 (1989) 562–580.
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Fukushima, M. A successive quadratic programming method for a class of constrained nonsmooth optimization problems. Mathematical Programming 49, 231–251 (1990). https://doi.org/10.1007/BF01588789
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DOI: https://doi.org/10.1007/BF01588789