Exact Penalty Functions with Multidimensional Penalty Parameter and Adaptive Penalty Updates

We present a general theory of exact penalty functions with vectorial (multidimensional) penalty parameter for optimization problems in infinite dimensional spaces. In comparison with the scalar case, the use of vectorial penalty parameters provides much more flexibility, allows one to adaptively and independently take into account the violation of each constraint during an optimization process, and often leads to a better overall performance of an optimization method using an exact penalty function. We obtain sufficient conditions for the local and global exactness of penalty functions with vectorial penalty parameters and study convergence of global exact penalty methods with several different penalty updating strategies. In particular, we present a new algorithmic approach to an analysis of the global exactness of penalty functions, which contains a novel characterisation of the global exactness property in terms of behaviour of sequences generated by certain optimization methods.Comment: In the second version, a number of small mistakes found in the paper was correcte

Firefly Penalty-based Algorithm for Bound Constrained Mixed-Integer Nonlinear Programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixedinteger nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favorably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed

An approach to constrained global optimization based on exact penalty functions

In the field of global optimization many efforts have been devoted to solve unconstrained global optimization problems. The aim of this paper is to show that unconstrained global optimization methods can be used also for solving constrained optimization problems, by resorting to an exact penalty approach. In particular, we make use of a non-differentiable exact penalty function . We show that, under weak assumptions, there exists a threshold value of the penalty parameter such that, for any , any global minimizer of P (q) is a global solution of the related constrained problem and conversely. On these bases, we describe an algorithm that, by combining an unconstrained global minimization technique for minimizing P (q) for given values of the penalty parameter and an automatic updating of that occurs only a finite number of times, produces a sequence {x (k) } such that any limit point of the sequence is a global solution of the related constrained problem. In the algorithm any efficient unconstrained global minimization technique can be used. In particular, we adopt an improved version of the DIRECT algorithm. Some numerical experimentation confirms the effectiveness of the approach