Combining approximation and exact penalty in hierarchical programming

The bundle modification strategy for the convex unconstrained problems was proposed by Alexey et al. [[2007] European Journal of Operation Research, 180(1), 38–47.] whose most interesting feature was the reduction of the calls for the quadratic programming solver. In this paper, we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems. Concretely, we adopt the convexification technique to the objective function and constraint function, take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies. The global convergence of the corresponding algorithm is proved. The primal numerical results show that the proposed algorithms are promising and effective.

Download Full-text

Exact penalty results for mathematical programs with vanishing constraints

Nonlinear Analysis ◽

10.1016/j.na.2009.10.047 ◽

2010 ◽

Vol 72 (5) ◽

pp. 2514-2526 ◽

Cited By ~ 30

Author(s):

Tim Hoheisel ◽

Christian Kanzow ◽

Jiří V. Outrata

Keyword(s):

Exact Penalty ◽

Mathematical Programs ◽

Vanishing Constraints

Download Full-text

Exact Penalization in Stochastic Programming—Calmness and Constraint Qualification

Advances in Decision Sciences ◽

10.1155/2014/569458 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Martin Branda

Keyword(s):

Stochastic Programming ◽

Error Bound ◽

Constraint Qualification ◽

Sufficient Conditions ◽

Discrete Distributions ◽

Exact Penalty ◽

Asymptotic Equivalence ◽

Exact Penalization ◽

Local Minimizers ◽

Constrained Problems

We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

Download Full-text

Smoothing Approximation to the Square-Root Exact Penalty Function

Journal of Systems Science and Information ◽

10.1515/jssi-2016-0087 ◽

2016 ◽

Vol 4 (1) ◽

pp. 87-96

Author(s):

Yaqiong Duan ◽

Shujun Lian

Keyword(s):

Penalty Function ◽

Original Problem ◽

Optimal Solution ◽

Penalty Functions ◽

Exact Penalty Functions ◽

Exact Penalty ◽

Square Root ◽

Inequality Constrained Optimization ◽

Numerical Examples ◽

Mild Conditions

AbstractIn this paper, smoothing approximation to the square-root exact penalty functions is devised for inequality constrained optimization. It is shown that an approximately optimal solution of the smoothed penalty problem is an approximately optimal solution of the original problem. An algorithm based on the new smoothed penalty functions is proposed and shown to be convergent under mild conditions. Three numerical examples show that the algorithm is efficient.

Download Full-text