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In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. We defne a linesearch-based solution method, and we show that it converges to a set of Pareto stationary points. To this aim, we carry out a theoretical analysis of the problem by only assuming Lipschitz continuity of the functions; more specifically, we give new optimality conditions that take explicitly into account the bound constraints, and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function. Finally, we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems, showing that our approach is promising when compared to a state-of-the-art method from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose new linesearch-based methods for nonsmooth constrained optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence toward stationary points, using the Clarke-Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the bound-constrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the bound-constrained case to solve the penalized problem. Moreover, we prove that every accumulation point, under standard assumptions on the search directions, of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both bound-constrained and nonlinearly constrained problems, showing that our approach is promising when compared to some state-of-the-art codes from the literature. [ABSTRACT FROM AUTHOR]

Mixed integer optimization is a powerful tool for modeling many optimization problems arising from real-world applications. Finding a first feasible solution represents the first step for several mixed integer programming (MIP) solvers. The feasibility pump is a heuristic for finding feasible solutions to mixed integer linear programming (MILP) problems which is effective even when dealing with hard MIP instances. In this work, we start by interpreting the feasibility pump as a Frank--Wolfe method applied to a nonsmooth concave merit function. Then we define a general class of functions that can be included in the feasibility pump scheme for measuring solution integrality, and we identify some merit functions belonging to this class. We further extend our approach by dynamically combining two different merit functions. Finally, we define a new version of the feasibility pump algorithm, which includes the original version of the feasibility pump as a special case, and we present computational results on binary MILP problems showing the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we propose a new derivative-free algorithm for linearly constrained finite minimax problems. Due to the nonsmoothness of this class of problems, standard derivative-free algorithms can locate only points which satisfy weak necessary optimality conditions. In this work we define a new derivative-free algorithm which is globally convergent toward standard stationary points of the finite minimax problem. To this end, we convert the original problem into a smooth one by using a smoothing technique based on the exponential penalty function of Kort and Bertsekas. This technique depends on a smoothing parameter which controls the approximation to the finite minimax problem. The proposed method is based on a sampling of the smooth function along a suitable search direction and on a particular updating rule for the smoothing parameter that depends on the sampling stepsize. Numerical results on a set of standard minimax test problems are reported. [ABSTRACT FROM AUTHOR]

In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we define a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573--594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]