An interval algorithm for nondifferentiable global optimization

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

In this paper, we study a new exact and smooth penalty function forsemi-infinite programming problems with continuous inequalityconstraints. Through this exact penalty function, we can transform asemi-infinite programming problem into an unconstrained optimizationproblem. We find that, under some reasonable conditions when thepenalty parameter is sufficiently large, the local minimizer of thispenalty function is the local minimizer of the primal problem.Moreover, under some mild assumptions, the local exactness propertyis explored. The numerical results demonstrate that it is aneffective and promising approach for solving constrainedsemi-infinite programming problems.

In many engineering optimization problems, the objective and the constraints which come from complex analytical models are often black-box functions with extensive computational effort. In this case, it is necessary for optimization process to use sampling data to fit surrogate models so as to reduce the number of objective and constraint evaluations as soon as possible. In addition, it is sometimes difficult for the constrained optimization problems based on surrogate models to find a feasible point, which is the premise of further searching for a global optimal feasible solution. For this purpose, a new Kriging-based Constrained Global Optimization (KCGO) algorithm is proposed. Unlike previous Kriging-based methods, this algorithm can dispose black-box constrained optimization problem even if all initial sampling points are infeasible. There are two pivotal phases in KCGO algorithm. The main task of the first phase is to find a feasible point when there is no feasible data in the initial sample. And the aim of the second phase is to obtain a better feasible point under the circumstances of fewer expensive function evaluations. Several numerical problems and three design problems are tested to illustrate the feasibility, stability and effectiveness of the proposed method.

Computationally expensive constrained optimization problems are challenging owing to their high complexity and computational cost. To solve these problems efficiently, a kriging-assisted bi-objective constrained global optimization (BOCGO) algorithm is developed, where three phases with three bi-objective subproblems are performed. In phase I, the constraints are searched locally and globally to find the feasible region. Once a feasible region has been located, the two terms of the constrained expected improvement function are utilized to exploit and explore the feasible region in phase II. As the kriging models are accurate enough in the concerned region, a local search is processed to improve the optimal solution in phase III. The capability of the BOCGO algorithm is demonstrated by comparison with two classical and two state-of-the-art algorithms on 20 problems and an engineering simulation problem. The results show that the BOCGO algorithm performs better in more than three-fifths of problems, illustrating its effectiveness and robustness.

We propose a Gauss–Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by Andre and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of the KKT conditions as a system of equations. Such reformulation allows the use of a semismooth Newton method, so that local superlinear convergence rate can be proved under an assumption weaker than the usual strong second-order sufficient condition and without requiring strict complementarity. Besides, we note that the exact penalty function can be used to globalize the method. We conclude with some numerical experiments using the collection of test problems CUTE.

An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive search algorithms.

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

In this paper, we will study optimality conditions of semi-infinite programs and generalized semi-infinite programs by employing lower order exact penalty functions and the condition that the generalized second-order directional derivative of the constraint function at the candidate point along any feasible direction for the linearized constraint set is non-positive. We consider three types of penalty functions for semi-infinite program and investigate the relationship among the exactness of these penalty functions. We employ lower order integral exact penalty functions and the second-order generalized derivative of the constraint function to establish optimality conditions for semi-infinite programs. We adopt the exact penalty function technique in terms of a classical augmented Lagrangian function for the lower-level problems of generalized semi-infinite programs to transform them into standard semi-infinite programs and then apply our results for semi-infinite programs to derive the optimality condition for generalized semi-infinite programs. We will give various examples to illustrate our results and assumptions.

<p style='text-indent:20px;'>In this work, an exact objective penalty function and an exact objective filled penalty function are proposed to solve constrained optimization problems. There are two main innovations of these two functions: using the exact objective penalty function to find a locally optimal point; a better locally optimal point than the current one can be found by exact objective filled penalty function. A local search method and a global search method for constrained optimization problems are proposed. We can use this new global search method to compute an approximately globally optimal point for constrained optimization problems. The convergence of the sequences obtained by the new algorithms are also analyzed respectively. Finally, the rationality of these two search methods are illustrated by numerical experiments.</p>

In order to improve the solution accuracy of the lightning attachment algorithm in dealing with constrained optimization problem, an improved constrained lightning attachment algorithm is proposed in this paper. Firstly, the parameter setting of the constraint processing technology was improved, and the constraint violation degree of all individuals in the population was added to further distinguish between feasible individuals and infeasible individuals; Secondly, several stages in the lightning attachment procedure optimization algorithm are improved, including the performance enhancement stage, the uplink leader stage, the downlink leader stage and so on. The improved constrained lightning attachment procedure optimization algorithm proposed in this paper is compared with the other three constrained optimization algorithms on the standard test function g01-g13. The results show that the algorithm proposed in this paper has certain advantages in solving precision.

An implementable algorithm for solving nonsmooth nonconvex constrained optimization is proposed by combining bundle ideas, proximity control, and the exact penalty function. We construct two kinds of approximations to nonconvex objective function; these two approximations correspond to the convex and concave behaviors of the objective function at the current point, which captures precisely the characteristic of the objective function. The penalty coefficients are increased only a finite number of times under the conditions of Slater constraint qualification and the boundedness of the constrained set, which limit the unnecessary penalty growth. The given algorithm converges to an approximate stationary point of the exact penalty function for constrained nonconvex optimization with weakly semismooth objective function. We also provide the results of some preliminary numerical testing to show the validity and efficiency of the proposed method.

In this paper we consider inequality constrained nonlinear optimization problems where the first order derivatives of the objective function and the constraints cannot be used. Our starting point is the possibility to transform the original constrained problem into an unconstrained or linearly constrained minimization of a nonsmooth exact penalty function. This approach shows two main difficulties: the first one is the nonsmoothness of this class of exact penalty functions which may cause derivative-free codes to converge to nonstationary points of the problem; the second one is the fact that the equivalence between stationary points of the constrained problem and those of the exact penalty function can only be stated when the penalty parameter is smaller than a threshold value which is not known a priori. In this paper we propose a derivative-free algorithm which overcomes the preceding difficulties and produces a sequence of points that admits a subsequence converging to a Karush–Kuhn–Tucker point of the constrained problem. In particular the proposed algorithm is based on a smoothing of the nondifferentiable exact penalty function and includes an updating rule which, after at most a finite number of updates, is able to determine a “right value” for the penalty parameter. Furthermore we present the results obtained on a real world problem concerning the estimation of parameters in an insulin-glucose model of the human body.

ABSTRACTIn this article, the vector exact l1 penalty function method used for solving nonconvex nondifferentiable multiobjective programming problems is analyzed. In this method, the vector penalized optimization problem with the vector exact l1 penalty function is defined. Conditions are given guaranteeing the equivalence of the sets of (weak) Pareto optimal solutions of the considered nondifferentiable multiobjective programming problem and of the associated vector penalized optimization problem with the vector exact l1 penalty function. This equivalence is established for nondifferentiable invex vector optimization problems. Some examples of vector optimization problems are presented to illustrate the results established in the article.

This paper presents the use of a constraint-handling technique, known as feasibility and dominance rules, in a electromagnetism-like (ELM) mechanism for solving constrained global optimization problems. Since the original ELM algorithm is specifically designed for solving bound constrained problems, only the inequality and equality constraints violation together with the objective function value are used to select points and to progress towards feasibility and optimality. Numerical experiments are presented, including a comparison with other methods recently reported in the literature.

In this chapter, an implementation of an interior point filter line-search algorithm for large-scale nonlinear programming proposed by Wachter and Biegler (2005a, b) is presented. As we know, to allow convergence from poor starting points and to enforce progress to the solution, interior point methods both in trust region and in line-search frameworks with exact penalty merit function have been developed. For example, KNITRO uses the l 1 exact penalty function (Byrd, Hribar, & Nocedal, 1999; Byrd, Gilbert, & Nocedal, 2000). On the other hand, Fletcher and Leyffer (2002) proposed filter methods as an alternative to merit functions, as a tool for global convergence guarantee in algorithms for nonlinear optimization. The idea of filter is that trial points are accepted if they improve the objective function value or improve the constraint violation instead of a combination of these two measures defined by the merit function. Even if the filter methods include different heuristics, they have been adapted to barrier methods in a number of ways. For example, Ulbrich et al. (2004) considered a trust-region filter method and accept the trial step on the basis of the norm of the optimality conditions. Benson et al. (2002a) proposed several heuristics using the idea of filter methods, for which improved efficiency is reported compared to their previous merit function approach. The global convergence of an interior point algorithm with a filter line search was analyzed by Wachter and Biegler (2001). Here, the assumptions made for the analysis of the global convergence are less restrictive than those made for line-search interior point methods for nonlinear programming developed, for example, by El-Bakry et al. (1996), Yamashita (1998), or Tits et al. (2003).