Duality and penalization in optimization via an augmented Lagrangian function with applications. (English) Zbl 1182.90097
To establish good candidates that an augmented Lagrangian functions reaches a zero duality gap, is a nice challenge in optimization, This paper provides an instance where it is possible to reach such goals.
After introducing the concepts of a valley at 0 augmenting function and the corresponding augmented Lagrangian problem, it is established the existence of solutions of a primal problem and its valley at 0 augmented Lagrangian problem in a reflexive Banach space. Then, the zero duality gap property is stated between the primal problem and the valley at 0 augmented Lagrangian dual problem.
Finally, the results are applied to an inequality and equality constrained optimization problem in infinite-dimensional spaces and variational problems in Sobolev spaces.
After introducing the concepts of a valley at 0 augmenting function and the corresponding augmented Lagrangian problem, it is established the existence of solutions of a primal problem and its valley at 0 augmented Lagrangian problem in a reflexive Banach space. Then, the zero duality gap property is stated between the primal problem and the valley at 0 augmented Lagrangian dual problem.
Finally, the results are applied to an inequality and equality constrained optimization problem in infinite-dimensional spaces and variational problems in Sobolev spaces.
Reviewer: Fabián Flores-Bazan (Concepción)
MSC:
| 90C48 | Programming in abstract spaces |
Keywords:
valley-at-0 augmented Lagrangian function; zero duality gap; exact penalty function; reflexive Banach spaceReferences:
| [1] | Cea, J.: Lectures on Optimization-Theory and Algorithms. Springer, Berlin (1978) · Zbl 0409.90050 |
| [2] | Luenberger, D.G.: Linear and Nonlinear Programming. Kluwer Academic, Berlin (1984) |
| [3] | Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969) · Zbl 0174.20705 · doi:10.1007/BF00927673 |
| [4] | Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974) · doi:10.1137/0312021 |
| [5] | Rockafellar, R.T.: Lagrange multipliers and optimality. SIAM Rev. 35, 183–238 (1993) · Zbl 0779.49024 · doi:10.1137/1035044 |
| [6] | Bertsekas, D.P.: Constrained Optimization and Lagrangian Multiplier Methods. Academic, New York (1982) · Zbl 0572.90067 |
| [7] | Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998) |
| [8] | Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28, 533–552 (2003) · Zbl 1082.90135 · doi:10.1287/moor.28.3.533.16395 |
| [9] | Rubinov, A.M., Huang, X.X., Yang, X.Q.: The zero duality gap property and lower semicontinuity of the perturbation function. Math. Oper. Res. 27, 775–791 (2002) · Zbl 1082.90567 · doi:10.1287/moor.27.4.775.295 |
| [10] | Zhou, Y.Y., Yang, X.Q.: Some results about duality and exact penalization. J. Glob. Optim. 29, 497–509 (2004) · Zbl 1075.90077 · doi:10.1023/B:JOGO.0000047916.73871.88 |
| [11] | Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991) · Zbl 0737.90060 · doi:10.1137/0329054 |
| [12] | Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 |
| [13] | Rubinov, A.M., Glover, B.M., Yang, X.Q.: Decreasing functions with applications to penalization. SIAM J. Optim. 10, 289–313 (1999) · Zbl 0955.90127 · doi:10.1137/S1052623497326095 |
| [14] | Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976) · Zbl 0404.90100 · doi:10.1287/moor.1.2.165 |
| [15] | Ito, K., Kunisch, K.: Augmented Lagrangian-SQP-method in Hilbert space and application to control in the coefficients problems. SIAM J. Optim. 6, 96–125 (1996) · Zbl 0846.65026 · doi:10.1137/0806007 |
| [16] | Forsyth, P.A., Vetzal, K.R.: Quadratic convergence for valuing American options using a penalty method. SIAM J. Sci. Comput. 23, 2095–2122 (2002) · Zbl 1020.91017 · doi:10.1137/S1064827500382324 |
| [17] | Penot, J.P.: Augmented Lagrangians, duality and growth conditions. J. Nonlinear Convex Anal. 3, 283–302 (2002) · Zbl 1030.49035 |
| [18] | Rubinov, A.M., Yang, X.Q.: Lagrange-Type Functions in Constrained Non-convex Optimization. Kluwer Academic, Berlin (2003) · Zbl 1049.90066 |
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