## Decreasing Functions With Applications To Penalization (1997)

Venue: | SIAM J. Optimization |

Citations: | 10 - 5 self |

### BibTeX

@ARTICLE{Rubinov97decreasingfunctions,

author = {A. M. Rubinov and B. M. Glover and X. Q. Yang},

title = {Decreasing Functions With Applications To Penalization},

journal = {SIAM J. Optimization},

year = {1997},

volume = {10},

pages = {289--313}

}

### OpenURL

### Abstract

: The theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions. A multiplicative version of the inf-convolution operation is studied for decreasing functions. Modified penalty functions for some constrained optimization problems are introduced which are in general nonlinear with respect to the objective function of the original problem. As the perturbation function of a constrained optimization problem is decreasing, the theory of decreasing functions is subsequently applied to the study of modified penalty functions, the zero duality gap property and the exact penalization. Key Words: Decreasing functions, IPH functions, multiplicative inf-convolution, modified penalty functions, exact penalization. AMS Subject Classification 1991: 90C30, 65K05 Abbreviated title: Decreasing functions and penalization 1 Introduction In this paper we study positive decreasing functions defined on the positive orthant IR n ...

### Citations

1156 |
Optimization and Nonsmooth Analysis
- Clarke
- 1983
(Show Context)
Citation Context ...= 1; : : : ; mg; y = (y 1 ; : : : y m ); has useful applications in the study of the nonlinear programming problem: f o (x) \Gamma! inf subject to x 2 X; f i (x)s0; i = 1; : : : ; m; (see for example =-=[3, 5, 6, 10, 11, 14]-=- and the references therein). The perturbation function is decreasing : y 1sy 2 =) fi(y 1 )sfi(y 2 ). Consequently the study of perturbation functions should be based on a theory of decreasing functio... |

562 |
Constrained optimization and Lagrange multiplier methods
- Bertsekas
- 1982
(Show Context)
Citation Context ...= 1; : : : ; mg; y = (y 1 ; : : : y m ); has useful applications in the study of the nonlinear programming problem: f o (x) \Gamma! inf subject to x 2 X; f i (x)s0; i = 1; : : : ; m; (see for example =-=[3, 5, 6, 10, 11, 14]-=- and the references therein). The perturbation function is decreasing : y 1sy 2 =) fi(y 1 )sfi(y 2 ). Consequently the study of perturbation functions should be based on a theory of decreasing functio... |

146 |
Optimization Theory for Large Systems
- Lasdon
- 1970
(Show Context)
Citation Context ...= 1; : : : ; mg; y = (y 1 ; : : : y m ); has useful applications in the study of the nonlinear programming problem: f o (x) \Gamma! inf subject to x 2 X; f i (x)s0; i = 1; : : : ; m; (see for example =-=[3, 5, 6, 10, 11, 14]-=- and the references therein). The perturbation function is decreasing : y 1sy 2 =) fi(y 1 )sfi(y 2 ). Consequently the study of perturbation functions should be based on a theory of decreasing functio... |

98 | Lagrange multipliers and optimality
- Rockafellar
- 1993
(Show Context)
Citation Context |

66 | Smoothing methods for convex inequalities and linear complementarity problems
- Chen, Mangasarian
- 1995
(Show Context)
Citation Context ...nvolution functions p such that the penalty function L + p is smooth at the solution. The corresponding question for the classical function (except the question 3)) have been discussed for example in =-=[2, 5, 7, 11, 14]-=-. 2 The first three questions are addressed in this paper. We consider the penalty function of the form (1.1) involving only an IPH function p with some natural properties. It is demonstrated in the p... |

42 |
Programmation mathématique, théorie et algorithmes: Tome1. C.N.E.T et E.N.S.T
- Minoux
- 1983
(Show Context)
Citation Context |

28 |
An exact penalization viewpoint of constrained optimization
- Burke
- 1991
(Show Context)
Citation Context |

22 |
Calmness and exact penalization
- Burke
- 1991
(Show Context)
Citation Context ... follows from Propositions 2.1 and 2.2 that s(r p ) = " x2X 1 s(q x p ) = " x2X 1 (f o (x); f 1 (x)) \Delta s(p): 2 17 6 Perturbation functions We now study the perturbation function (see fo=-=r example [4, 5, 6, 14, 13]-=- and references therein) for the problem (P ) defined by (5.1). Definition 6.1 The function fi defined on IR I ++ by fi(y) = infff o (x) : x 2 X; f 1 (x)syg (y ? 0) (6.1) is called the perturbation fu... |

17 |
Minkowski duality and its applications
- Kutateladze, Rubinov
- 1972
(Show Context)
Citation Context ...degree (IPH functions). There exists a natural isomorphism between the ordered space of all positive decreasing functions and the ordered space of all IPH functions. Theory of abstract convexity (see =-=[9, 12, 18]-=-) allows us to consider duality 1 This research has been supported by grants from the Australian Research Council 1 in various nonconvex situations. Recently [15] a duality theory based on abstract co... |

11 |
Duality for increasing positively homogeneous functions and normal sets, to appear
- Rubinov, Glover
- 1998
(Show Context)
Citation Context ...ory of abstract convexity (see [9, 12, 18]) allows us to consider duality 1 This research has been supported by grants from the Australian Research Council 1 in various nonconvex situations. Recently =-=[15]-=- a duality theory based on abstract convexity with respect to the so-called min-type functions was developed for IPH functions. The isomorphism between IPH functions and decreasing functions allows us... |

8 |
Asymptotic analysis for penalty and barrier methods in convex and linear programming
- Auslender, Cominetti, et al.
- 1997
(Show Context)
Citation Context ...nvolution functions p such that the penalty function L + p is smooth at the solution. The corresponding question for the classical function (except the question 3)) have been discussed for example in =-=[2, 5, 7, 11, 14]-=-. 2 The first three questions are addressed in this paper. We consider the penalty function of the form (1.1) involving only an IPH function p with some natural properties. It is demonstrated in the p... |

4 |
Abstract Convex Analysis, Wiley-Interscience Publication
- Singer
- 1997
(Show Context)
Citation Context ...degree (IPH functions). There exists a natural isomorphism between the ordered space of all positive decreasing functions and the ordered space of all IPH functions. Theory of abstract convexity (see =-=[9, 12, 18]-=-) allows us to consider duality 1 This research has been supported by grants from the Australian Research Council 1 in various nonconvex situations. Recently [15] a duality theory based on abstract co... |

3 | A su#cient and necessary condition for nonconvex constrained optimization - Goh, Yang - 1997 |

3 |
Modified Lagrange and penalty functions in continuous optimization
- Rubinov, Glover, et al.
- 1999
(Show Context)
Citation Context ...ion p(y 1 ; y 2 ) = y 1 + y 2 . Sometimes it is more convenient consider the convolution by an increasing function p with some additional properties. This approach has been developed by Rubinov at al =-=[17]-=- and Andramonov [1]. In such a case the modified penalty function L + p (x; d) has the form L + p (x; d) = p(f o (x); d max(f 1 (x); 0)) (1.1) Among the many questions which arise in connection with s... |

3 |
Exact auxiliary functions in optimization problems
- Yevtushenko, Zhadan
- 1990
(Show Context)
Citation Context ...y 2 ). Consequently the study of perturbation functions should be based on a theory of decreasing functions. It is well known that all constraints f i can be convoluted into a single constraint. (See =-=[19]-=- for a detailed discussion.) For example we can use a convolution by maximum: max i f i (x)s0 () f i (x)s0 for all i. So we restrict ourselves to the problem with a single constraint. The classical pe... |

2 | Some properties of increasing convex-along-rays functions (submitted paper
- Rubinov
- 1998
(Show Context)
Citation Context ...y 2 IR I ++ g (2.2) is called the support set of the function p. For IPH functions defined on the cone IR I + it is possible to define a support set in different ways. One of them has been studied in =-=[16]-=-. However it will be more convenient to use the following definition in this paper. Definition 2.2 Letsp be an IPH function defined on IR I + and let p be the restriction of the functionsp to the cone... |

1 |
Andramonov, An approach to constructing generalized penalty functions
- Yu
- 1997
(Show Context)
Citation Context ...y 1 + y 2 . Sometimes it is more convenient consider the convolution by an increasing function p with some additional properties. This approach has been developed by Rubinov at al [17] and Andramonov =-=[1]-=-. In such a case the modified penalty function L + p (x; d) has the form L + p (x; d) = p(f o (x); d max(f 1 (x); 0)) (1.1) Among the many questions which arise in connection with such a setting we in... |

1 |
Conjugate Duality and Optimization, No 16
- Rockafellar
- 1974
(Show Context)
Citation Context ... follows from Propositions 2.1 and 2.2 that s(r p ) = " x2X 1 s(q x p ) = " x2X 1 (f o (x); f 1 (x)) \Delta s(p): 2 17 6 Perturbation functions We now study the perturbation function (see fo=-=r example [4, 5, 6, 14, 13]-=- and references therein) for the problem (P ) defined by (5.1). Definition 6.1 The function fi defined on IR I ++ by fi(y) = infff o (x) : x 2 X; f 1 (x)syg (y ? 0) (6.1) is called the perturbation fu... |