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This paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available. We introduce the mesh adaptive direct search (MADS) class of algorithms which extends the generalized pattern search (GPS) class by allowing local exploration, called polling, in an asymptotically dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points x0302;, where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions span the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for GPS. The main result of this paper is that the general MADS framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the Clarke derivatives are nonnegative. We propose an instance of MADS for which the refining directions are dense in the hypertangent cone at x0302; with probability 1 whenever the iterates associated with the refining directions converge to a single x0302;. The instance of MADS is compared to versions of GPS on some test problems. We also illustrate the limitation of our results with examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we generalize the inclusion constrained longest common subsequence (CLCS) problem to the hybrid CLCS problem which is the combination of the sequence inclusion CLCS and the string inclusion CLCS, called the sequential substring constrained longest common subsequence (SSCLCS) problem. In the SSCLCS problem, we are given two strings and of lengths and , respectively, formed by alphabet and a constraint sequence formed by ordered strings with total length . The problem is that of finding the longest common subsequence of and containing as substrings and with the order of the ’s retained. This problem has two variants, depending on whether the strings in cannot overlap or may overlap. We propose algorithms with and time for the two variants. For the special case with one or two constraints, our algorithm runs in or time, respectively—an order faster than the algorithm proposed by Chen and Chao. [Copyright &y& Elsevier]

Abstract: In this paper, we consider numerical simulation to a class of constrained dynamic problems where the overall dynamics are determined by the interactions between two sub-systems. We present an iterative algorithm that naturally decouples the computation of the two sub-systems and that ensures an accurate and efficient solution procedure. We conduct rigorous error analysis for the convergence of the iterative algorithm, and verify the analytical results through careful numerical tests. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new penalty-free-type method for nonlinear equality constrained problems. The new algorithm uses trust region framework and feasibility safeguarding technique. Moreover, it has no choice of penalty parameter and penalty function as a merit function, and it does not use the filter technique to avoid the penalty function either. We analyze the global convergence of the main algorithm under the standard assumptions. The preliminary numerical tests are reported. [Copyright &y& Elsevier]

Abstract: In this paper, an active set limited BFGS algorithm is proposed for bound constrained optimization. The global convergence will be established under some suitable conditions. Numerical results show that the given method is effective. [Copyright &y& Elsevier]

Abstract: A directed searching optimization algorithm (DSO) is proposed to solve constrained optimization problems in this paper. The proposed algorithm includes two important operations — position updating and genetic mutation. Position updating enables the non-best solution vectors to mimic the best one, which is beneficial to the convergence of the DSO; genetic mutation can increase the diversity of individuals, which is beneficial to preventing the premature convergence of the DSO. In addition, we adopt the penalty function method to balance objective and constraint violations. We can obtain satisfactory solutions for constrained optimization problems by combining the DSO and the penalty function method. Experimental results indicate that the proposed algorithm can be an efficient alternative on solving constrained optimization problems. [Copyright &y& Elsevier]

Abstract: A novel modified differential evolution algorithm (NMDE) is proposed to solve constrained optimization problems in this paper. The NMDE algorithm modifies scale factor and crossover rate using an adaptive strategy. For any solution, if it is at a standstill, its own scale factor and crossover rate will be adjusted in terms of the information of all successful solutions. We can obtain satisfactory feasible solutions for constrained optimization problems by combining the NMDE algorithm and a common penalty function method. Experimental results show that the proposed algorithm can yield better solutions than those reported in the literature for most problems, and it can be an efficient alternative to solving constrained optimization problems. [Copyright &y& Elsevier]

Abstract: In this paper, a nonmonotone trust region algorithm for unconstrained optimization problems is presented. In the algorithm, a kind of nonmonotone technique, which is evidently different from Grippo, Lampariello and Lucidi’s approach, is used. Under mild conditions, global and local convergence results of the algorithm are established. Preliminary numerical results show that the new algorithm is efficient. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to incorporate the preconditioned gradient path in a nonmonotone stabilization algorithm for unconstrained optimization. The global convergence and locally superlinear convergence are established for this class of algorithms. Finally, we report in details the numerical results which show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new branch-and-bound algorithm for the general quadratic problems with box constraints. We, first, transform the problem into a separable form by D. C. decomposition and Cholesky factorization of a positive definite matrix. Then a lower bounding technique is derived and a branch-and-bound algorithm is presented based on the lower bounding and rectangular bisection. Finally, preliminary computational results are reported. [ABSTRACT FROM AUTHOR]

We say that $X=[x_{ij}]_{i,j=1}^n$ is symmetric centrosymmetric if x ij = x ji and x n − j + 1, n − i + 1, 1 ≤ i, j ≤ n. In this paper we present an efficient algorithm for minimizing || AXA T − B|| where ||·|| is the Frobenius norm, A ∈ ℝ m× n, B ∈ ℝ m× m and X ∈ ℝ n× n is symmetric centrosymmetric with a specified central submatrix [ x ij] p ≤ i, j ≤ n − p. Our algorithm produces a suitable X such that AXA T = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T. We use the non-convex constraint $${ Y - x x^T \preccurlyeq 0}$$ to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint $${ Y - x x^T \succcurlyeq 0}$$ to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings. [ABSTRACT FROM AUTHOR]

This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a real coded genetic algorithm named MI-LXPM is proposed for solving integer and mixed integer constrained optimization problems. The proposed algorithm is a suitably modified and extended version of the real coded genetic algorithm, LXPM, of Deep and Thakur [K. Deep, M. Thakur, A new crossover operator for real coded genetic algorithms, Applied Mathematics and Computation 188 (2007) 895–912; K. Deep, M. Thakur, A new mutation operator for real coded genetic algorithms, Applied Mathematics and Computation 193 (2007) 211–230]. The algorithm incorporates a special truncation procedure to handle integer restrictions on decision variables along with a parameter free penalty approach for handling constraints. Performance of the algorithm is tested on a set of twenty test problems selected from different sources in literature, and compared with the performance of an earlier application of genetic algorithm and also with random search based algorithm, RST2ANU, incorporating annealing concept. The proposed MI-LXPM outperforms both the algorithms in most of the cases which are considered. [Copyright &y& Elsevier]

This paper proposes an adaptive penalty function for solving constrained optimization problems using genetic algorithms. The proposed method aims to exploit infeasible individuals with low objective value and low constraint violation. The number of feasible individuals in the population is used to guide the search process either toward finding more feasible individuals or searching for the optimum solution. The proposed method is simple to implement and does not need any parameter tuning. The performance of the algorithm is tested on 22 benchmark functions in the literature. The results show that the proposed approach is able to find very good solutions comparable to the chosen state-of-the-art designs. Furthermore, it is able to find feasible solutions in every run for all of the benchmark functions tested. [ABSTRACT FROM AUTHOR]

Abstract: In image restoration, the so-called edge-preserving regularization method is used to solve an optimization problem whose objective function has a data fidelity term and a regularization term, the two terms are balanced by a parameter . In some aspect, the value of determines the quality of images. In this paper, we establish a new model to estimate the parameter and propose an algorithm to solve the problem. In order to improve the quality of images, in our algorithm, an image is divided into some blocks. On each block, a corresponding value of has to be determined. Numerical experiments are reported which show efficiency of our method. [Copyright &y& Elsevier]

In this paper we discuss the polynomiality of a feasible version of Mehrotra's predictor-corrector algorithm whose variants have been widely used in several interior point method (IPM)-based optimization packages. A numerical example is given that shows that the adaptive choice of centering parameter and correction terms in this algorithm may lead to small steps being taken in order to keep the iterates in a large neighborhood of the central path, which is important for proving polynomial complexity properties of this method. Motivated by this example, we introduce a safeguard in Mehrotra's algorithm that keeps the iterates in the prescribed neighborhood and allows us to obtain a positive lower bound on the step size. This safeguard strategy is also used when the affine scaling direction performs poorly. We prove that the safeguarded algorithm will terminate after at most O(n2log(x°}T s° /e) iterations. By modestly modifying the corrector direction, we reduce the iteration complexity to O(nlog(x°}T s° /e). To ensure fast asymptotic convergence of the algorithm, we changed Mehrotra's updating scheme of the centering parameter slightly while keeping the safeguard. The new algorithms have the same order of iteration complexity as the safeguarded algorithms but enjoy superlinear convergence as well. Numerical results using the McIPM and LIPSOL software packages are reported. [ABSTRACT FROM AUTHOR]

In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111-124 (2007)]. Three robust nonlinear programming problems were chosen from the literature using the criteria that results calculated using other methods must be available and the problems should be realistic, but fairly simple. Our studies show that the first-order method produced reasonable solutions when the level of uncertainty was small to moderate. In addition, we demonstrate a method for leveraging a theoretical result to eliminate constraint violations. Since the first-order method is relatively inexpensive in comparison to other robust optimization techniques, our studies indicate that, under moderate uncertainty, the first-order approach may be more suitable than other methods for large problems. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to propose and compare different one-dimensional maps as chaotic search patterns in the constraint nonlinear optimization problems. For this purpose, about 10 one-dimensional maps are introduced that can be used as search pattern in chaos optimization algorithms. We apply these maps in specific optimization algorithm (weighted gradient direction based chaos optimization algorithm) and compare them based on numerical simulation results. [Copyright &y& Elsevier]

Abstract: Many practical problems can be formulated as constrained optimization problems. Due to the simple concept and easy implementation, the penalty function method has been one of the most common techniques to handle constraints. However, the performance of this technique greatly relies on the setting of penalty factors, which are usually determined by manual trial and error, and the suitable penalty factors are often problem-dependent and difficult to set. In this paper, a differential evolution approach based on a co-evolution mechanism, named CDE, is proposed to solve the constrained problems. First, a special penalty function is designed to handle the constraints. Second, a co-evolution model is presented and differential evolution (DE) is employed to perform evolutionary search in spaces of both solutions and penalty factors. Thus, the solutions and penalty factors evolve interactively and self-adaptively, and both the satisfactory solutions and suitable penalty factors can be obtained simultaneously. Simulation results based on several benchmark functions and three well-known constrained design problems as well as comparisons with some existed methods demonstrate the effectiveness, efficiency and robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed, and a new SQP feasible descent algorithm for solving the problems is presented. At each iteration of the new algorithm, a convex quadratic program (QP) which always has feasible solution is solved and a master direction is obtained, then, an improved (feasible descent) direction is yielded by updating the master direction with an explicit formula, and in order to avoid the Maratos effect, a height-order correction direction is computed by another explicit formula of the master direction and the improved direction. The new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Furthermore, the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted. Finally, some numerical tests are reported. [Copyright &y& Elsevier]

In this paper, a class of deformable contour methods using a constrained optimization approach of minimizing a contour energy function satisfying an interior homogeneity constraint is proposed. The class is defined by any positive potential function describing the contour interior characterization. An evolutionary strategy is used to derive the algorithm. A similarity threshold Tv can be used to determine the interior size and shape of the contour. Sensitivity and significance of Tv and σ (a spreadness measure) are also discussed and shown. Experiments on noisy images and the convergence to a minimum energy gap contour are included. The developed method has been applied to a variety of medical images from CT abdominal section, MRI image slices of brain, brain tumor, a pig heart ultrasound image sequence to visual blood cell images. As the results show, the algorithm can be adapted to a broad range of medical images containing objects with vague, complex and/or irregular shape boundary, inhomogeneous and noisy interior, and contour with small gaps. [ABSTRACT FROM AUTHOR]

In [SIAM J. Optim., 17 (2006), pp. 188-217] Audet and Dennis proposed the class of mesh adaptive direct search (MADS) algorithms for minimization of a nonsmooth function under general nonsmooth constraints. The notation used in the paper evolved since the preliminary versions, and, unfortunately, even though the statement of Proposition 4.2 is correct, it is not compatible with the final notation. The purpose of this note is to show that the proposition is valid. [ABSTRACT FROM AUTHOR]