ABSTRACT We study the rate of convergence of a projected Barzilai-Borwein method, which performs ... more ABSTRACT We study the rate of convergence of a projected Barzilai-Borwein method, which performs the Grippo-Lampariello-Lucidi (GLL) non-monotone line search along the feasible direction, for convex constrained optimization. Under mild conditions, we establish sublinear convergence of the considered method for general convex functions. Moreover, we show that the rate of convergence is R-linear for strongly convex functions. We extend the convergence results to different versions of the method based on a general GLL non-monotone line search, an adaptive non-monotone line search, and a non-monotone line search that requires an average of the successive function values decreases.
ABSTRACT In this paper, we present an efficient method for nonnegative matrix factorization based... more ABSTRACT In this paper, we present an efficient method for nonnegative matrix factorization based on the alternating nonnegative least squares framework. Our approach adopts a monotone projected Barzilai–Borwein (MPBB) method as an essential subroutine where the step length is determined without line search. The Lipschitz constant of the gradient is exploited to accelerate convergence. Global convergence of the proposed MPBB method is established. Numerical results are reported to demonstrate the efficiency of our algorithm.
The paper is concerned with the expected residual minimization formulation (ERM) of the stochasti... more The paper is concerned with the expected residual minimization formulation (ERM) of the stochastic P matrix linear complementarity problem. It is shown that the expected residual minimization formulation of this problem has a nonempty and bounded solution set when the involving matrix is a stochastic P matrix. The definition and some properties of a stochastic P 0 matrix are given. Moreover, we consider the discrete case and give a condition for the solution set of the ERM problem to be singleton.
In this paper, based on the regularization techniques and projected gradient strategies, we prese... more In this paper, based on the regularization techniques and projected gradient strategies, we present a quadratic regularization projected alternating Barzilai–Borwein (QRPABB) method for minimizing differentiable functions on closed convex sets. We show the convergence of the QRPABB method to a constrained stationary point for a nonmonotone line search. When the objective function is convex, we prove the error in the objective function at iteration k is bounded by a k+1 for some a independent of k. Moreover, if the objective function is strongly convex, then the convergence rate is R-linear. Numerical comparisons of methods on box-constrained quadratic problems and nonnegative matrix factorization problems show that the QRPABB method is promising.
In this paper, the Newton-KKT interior-point method for indefinite quadratic programming is exten... more In this paper, the Newton-KKT interior-point method for indefinite quadratic programming is extended and applied to some special linear complementarity problems. Global and local quadratic convergence properties of the extended method are analyzed under nondegeneracy assumptions. Numerical results show that the proposed method is practical.
ABSTRACT We consider the expected residual minimization (ERM) formulation of stochastic linear co... more ABSTRACT We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.
International Journal of Computer Mathematics, 2012
ABSTRACT In this paper, we propose a new trust-region algorithm for bound-constrained semismooth ... more ABSTRACT In this paper, we propose a new trust-region algorithm for bound-constrained semismooth systems of equations. Trust-region subproblem is defined by minimizing a quadratic function subject only to a rectangular constraint. By employing a new active set and nonmonotone techniques, solution of the equations can be found effective. Global and local convergence results of the proposed algorithm are established under reasonable conditions. The algorithm is applied and tested on complementary problems and the experiments show that our method is efficient.
ABSTRACT A class of stochastic linear complementarity problems (SLCPs) with finitely many realiza... more ABSTRACT A class of stochastic linear complementarity problems (SLCPs) with finitely many realizations is considered. We first formulate the problem as a new constrained minimization problem. Then, we propose a feasible semismooth Newton method which yields a stationary point of the constrained minimization problem. We study the condition for the level set of the objective function to be bounded. As a result, the condition for the solution set of the constrained minimization problem is obtained. The global and quadratic convergence of the proposed method is proved under certain assumptions. Preliminary numerical results show that this method yields a reasonable solution with high safety and within a small number of iterations.
ABSTRACT We study the rate of convergence of a projected Barzilai-Borwein method, which performs ... more ABSTRACT We study the rate of convergence of a projected Barzilai-Borwein method, which performs the Grippo-Lampariello-Lucidi (GLL) non-monotone line search along the feasible direction, for convex constrained optimization. Under mild conditions, we establish sublinear convergence of the considered method for general convex functions. Moreover, we show that the rate of convergence is R-linear for strongly convex functions. We extend the convergence results to different versions of the method based on a general GLL non-monotone line search, an adaptive non-monotone line search, and a non-monotone line search that requires an average of the successive function values decreases.
ABSTRACT In this paper, we present an efficient method for nonnegative matrix factorization based... more ABSTRACT In this paper, we present an efficient method for nonnegative matrix factorization based on the alternating nonnegative least squares framework. Our approach adopts a monotone projected Barzilai–Borwein (MPBB) method as an essential subroutine where the step length is determined without line search. The Lipschitz constant of the gradient is exploited to accelerate convergence. Global convergence of the proposed MPBB method is established. Numerical results are reported to demonstrate the efficiency of our algorithm.
The paper is concerned with the expected residual minimization formulation (ERM) of the stochasti... more The paper is concerned with the expected residual minimization formulation (ERM) of the stochastic P matrix linear complementarity problem. It is shown that the expected residual minimization formulation of this problem has a nonempty and bounded solution set when the involving matrix is a stochastic P matrix. The definition and some properties of a stochastic P 0 matrix are given. Moreover, we consider the discrete case and give a condition for the solution set of the ERM problem to be singleton.
In this paper, based on the regularization techniques and projected gradient strategies, we prese... more In this paper, based on the regularization techniques and projected gradient strategies, we present a quadratic regularization projected alternating Barzilai–Borwein (QRPABB) method for minimizing differentiable functions on closed convex sets. We show the convergence of the QRPABB method to a constrained stationary point for a nonmonotone line search. When the objective function is convex, we prove the error in the objective function at iteration k is bounded by a k+1 for some a independent of k. Moreover, if the objective function is strongly convex, then the convergence rate is R-linear. Numerical comparisons of methods on box-constrained quadratic problems and nonnegative matrix factorization problems show that the QRPABB method is promising.
In this paper, the Newton-KKT interior-point method for indefinite quadratic programming is exten... more In this paper, the Newton-KKT interior-point method for indefinite quadratic programming is extended and applied to some special linear complementarity problems. Global and local quadratic convergence properties of the extended method are analyzed under nondegeneracy assumptions. Numerical results show that the proposed method is practical.
ABSTRACT We consider the expected residual minimization (ERM) formulation of stochastic linear co... more ABSTRACT We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai–Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.
International Journal of Computer Mathematics, 2012
ABSTRACT In this paper, we propose a new trust-region algorithm for bound-constrained semismooth ... more ABSTRACT In this paper, we propose a new trust-region algorithm for bound-constrained semismooth systems of equations. Trust-region subproblem is defined by minimizing a quadratic function subject only to a rectangular constraint. By employing a new active set and nonmonotone techniques, solution of the equations can be found effective. Global and local convergence results of the proposed algorithm are established under reasonable conditions. The algorithm is applied and tested on complementary problems and the experiments show that our method is efficient.
ABSTRACT A class of stochastic linear complementarity problems (SLCPs) with finitely many realiza... more ABSTRACT A class of stochastic linear complementarity problems (SLCPs) with finitely many realizations is considered. We first formulate the problem as a new constrained minimization problem. Then, we propose a feasible semismooth Newton method which yields a stationary point of the constrained minimization problem. We study the condition for the level set of the objective function to be bounded. As a result, the condition for the solution set of the constrained minimization problem is obtained. The global and quadratic convergence of the proposed method is proved under certain assumptions. Preliminary numerical results show that this method yields a reasonable solution with high safety and within a small number of iterations.
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