Abstract

A new class of smooth exact penalty functions, constructed using the hyperbolic tangent (tanh) function, is proposed for solving constrained global optimization problems. The tanh function is chosen because of its unique mathematical properties: it is monotonic, bounded, infinitely differentiable, and provides a uniform approximation of the absolute value function with an explicit error bound of O(1/(γe)). These properties make it particularly suitable for constructing smooth penalty functions that preserve exactness. The proposed penalty function exhibits both smoothness and exactness: it is continuously differentiable, and for a sufficiently large penalty parameter, its local minimizers coincide exactly with those of the original constrained problem. In addition, by integrating a filling term, a novel filled penalty function is constructed that enables the algorithm to escape from a current local minimizer and locate a better one. Leveraging this filled penalty function, a global optimization algorithm is designed that performs local minimization and filling stages alternately. The convergence properties of the algorithm are rigorously established; it is shown that the sequence of objective function values is strictly decreasing and that the termination point constitutes a global approximate optimal solution. Finally, numerical experiments on 18 benchmark problems, along with statistical significance tests and performance profiles, confirm the effectiveness and competitiveness of the proposed approach against traditional quadratic penalty methods and modern solvers such as IPOPT and ALM.

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