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In this paper, a modified BFGS algorithm is proposed for unconstrained optimization. The proposed algorithm has the following properties: (i) a nonmonotone line search technique is used to obtain the step size $\alpha_{k}$ to improve the effectiveness of the algorithm; (ii) the algorithm possesses not only global convergence but also superlinear convergence for generally convex functions; (iii) the algorithm produces better numerical results than those of the normal BFGS method. [ABSTRACT FROM AUTHOR]

In this paper, we present a flexible nonmonotone filter method for solving nonlinear constrained optimization problems which are common models in industry. This new method has more flexibility for the acceptance of the trial step compared to the traditional filter methods, and requires less computational costs compared with the monotone-type methods. Moreover, we use a self-adaptive parameter to adjust the acceptance criteria, so that Maratos effect can be avoided a certain degree. Under reasonable assumptions, the proposed algorithm is globally convergent. Numerical tests are presented that confirm the efficiency of the approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use a projected gradient algorithm to solve a nonlinear operator equation with -norm constraint. Gradient iterations with -norm constraints have been studied recently both in the context of inverse problem and of compressed sensing. In this paper, the constrained gradient iteration is implemented via a projected operator. We establish the -norm convergence of sequence constructed by the constrained gradient iteration when . The performance of the method is testified by a numerical example. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix. The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D. [Copyright &y& Elsevier]

Abstract: Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but reversal of the elimination principle can be beneficial as well: competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task, defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms, that is their convergence from the start. A well known example is the dual linear and nonlinear programming, which enhances the power of optimization algorithms. We believe that this is just an ad hoc application of general Principle of Expansion with Independent Constraints; it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization. At the end of this paper we comment on other applications and extensions of this principle. Presently we show it at work for the approximation of a single zero of a univariate polynomial of a degree . Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all zeros, such as Weierstrass–Durand–Kerner’s root-finder, which reduces its root-finding task to Viète’s (Vieta’s) system of polynomial equations with unknowns. We adjust this root-finder to the approximation of a single zero of , preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear. Together with computing a zero of a polynomial , the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of . Alternatively by using processors that exchange no data, one can concurrently approximate up to zeros of . Our tests confirm the efficiency of the proposed algorithms. Technically our root-finding boils down to computations with structured matrices, polynomials and partial fraction decompositions. Our study of these links can be of independent interest; e.g., as by-product we express the inverse of a Sylvester matrix via its last column, thus extending the celebrated result of Gohberg and Sementsul (1972) from Toeplitz to Sylvester matrix inverses. [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: This paper presents an augmented Lagrangian methodology with a stochastic population based algorithm for solving nonlinear constrained global optimization problems. The method approximately solves a sequence of simple bound global optimization subproblems using a fish swarm intelligent algorithm. A stochastic convergence analysis of the fish swarm iterative process is included. Numerical results with a benchmark set of problems are shown, including a comparison with other stochastic-type algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we presented a modified QP-free filter method based on a new piecewise linear NCP functions. In contrast with the existing QP-free methods, each iteration in this algorithm only needs to solve systems of linear equations which are derived from the equality part in the KKT first order optimality conditions. Its global convergence and local superlinear convergence are obtained under mild conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new feasible primal–dual interior point algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only two or three reduced systems of linear equations with the same coefficient matrix. The searching direction is feasible and the object function is monotone decreasing. The proposed algorithm is proved to possess global and superlinear convergence under mild conditions. Finally, some numerical experiments with the algorithm are reported. [Copyright &y& Elsevier]

Abstract: In this paper, by means of “working set” technique for determining the active set and the idea of primal–dual interior-point method, a new feasible QP-free algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only three reduced systems of linear equations with common coefficient matrix. Moreover, the initial iteration point can be at constraint boundary and the coefficient matrix is uniformly nonsingular without the strict complementarity. We also prove that the proposed algorithm obtains global and superlinear convergence. At last, preliminary numerical results are reported. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a kind of QP-free feasible algorithm which solves an inequality constrained nonlinear optimization problem. Under some weaker conditions than those in [H. Qi, L. Qi, A New QP-free, globally convergent, locally superlinear convergent algorithm for inequality constrained optimization, SIAM J. Optim. 11 (2000) 113–132], we prove that the algorithm is implementable and globally convergent. Moreover, some numerical test results are given to indicate that the algorithm is quite promising. [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]