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This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian matrix of the loss function, we establish an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, we establish its KL property of exponent 1/2 on the global optimal solution set under the noisy and full sample setting, and achieve this property at its certain class of critical points under the noisy and partial sample setting. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for solutions of perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set map at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in \cite{Hager99,Izmailov12} for nonlinear programming and in \cite{Cui16,Zhang17} for semidefinite programming., 21 pages

We utilize the spectral broadening of Yb-doped fiber lasers in the spectral beam combining scheme to develop an analytical model of the coupling efficiency, which forms a critical factor in evaluating the practicality of the beam combination system. The simulation results predict a trend similar to the measured ones. Via increasing the number of simulating lasers, the model can be extended to calculate the combining efficiency of the resulting multiple-beam-combination system and estimate the optimal output power and combining efficiency. Moreover, the analytical model is suitable to investigate key parameters of Yb-doped lasers and filters, which is beneficial in enhancing the combining efficiency.

We present a fast , easy and real-time method for measuring the laser beam quality factor, M2 , using a spatial light modulator(SLM). We import computer-generated holograms into the spatial light modulator. Then the laser beams are reflected by the SLM and transfer through a fourier transform lens, composed into different kinds of modes which are received by a CCD. The M2 value can be obtained by analyzing and calculating the modal weighting coefficients of different modes. In this paper, two different kinds of softwares , Matlab and VirtualLab, are used to implement the numerical simulations of the experiment process, so that the feasibility of the algorithm can be proved. In the end, we set up the experimental platform and implement the experiments with measuring laser beam quality factor, M2 , using a he-ne laser as the light source. The comparison with the theoretical value and the simulative value proves the reliability of the experiments.

The rank function rank(.) is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of rank(.), and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical results indicate that this convex relaxation method is comparable with the sequential semismooth Newton method (Li and Qi in SIAM J Optim 21:1641---1666, 2011) and the majorized penalty approach (Gao and Sun, 2010) in terms of the quality of solutions.

For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson's constraint qualification, we show that the nonsingularity of Clarke's Jacobian of the Fischer-Burmeister (FB) nonsmooth system is equivalent to the strong regularity of the Karush-Kuhn-Tucker point. Consequently, from Sun's paper [Math. Oper. Res., 31 (2006), pp. 761-776] the semismooth Newton method applied to the FB system may attain the locally quadratic convergence under the strong second order sufficient condition and constraint nondegeneracy.

We present a smooth approximation for the generalized Fischer–Burmeister function where the 2-norm in the FB function is relaxed to a general p -norm ( p > 1 ), and establish some favorable properties for it — for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ ( 1 , 2 ] has better numerical performance than the one with p > 2 .

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian R 01-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227–251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer–Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115–135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R 02 -property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293–327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer–Burmeister function, Math. Program. 107 (2006) 547–553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159–185] under a unified framework.

We consider a regularization method for nonlinear complementarity problems with F being a P"0-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer-Burmeister (FB) NCP-functions @f"p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example [email protected]?[1.1,2], usually has better numerical performance, and the generalized FB functions @f"p with [email protected]?[1.1,2) can be used as the substitutions for the FB function @f"2.

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.

Consider the problem of minimizing the sum of p-norms, where p is a fixed real number in the interval [1,2]. This nondifferentiable problem arises in many applications, including the VLSI (very-large-scale-integration) layout problem, the facilities location problem and the Steiner minimum tree problem under a given topology. In this paper, we establish the optimality conditions, duality and uniqueness results for the problem. We then present a smoothing Newton method by the semismooth equations which are derived from the optimality conditions. The method is globally and superlinearly convergent, and moreover, it is quadratically convergent when p∈[1,3/2]∪{2}. Particularly, the quadratic convergence is proved for the case wherep∈(1,3/2]∪{2} without requiring strict complementarity. Preliminary numerical results are reported, which indicate that the method proposed is extremely promising.

In this paper, we present a detailed investigation for the properties of a one-parametric class of SOC complementarity functions, which include the globally Lipschitz continuity, strong semismoothness, and the characterization of their B-subdifferential. Moreover, for the merit functions induced by them for the second-order cone complementarity problem (SOCCP), we provide a condition for each stationary point to be a solution of the SOCCP and establish the boundedness of their level sets, by exploiting Cartesian P-properties. We also propose a semismooth Newton type method based on the reformulation of the nonsmooth system of equations involving the class of SOC complementarity functions. The global and superlinear convergence results are obtained, and among others, the superlinear convergence is established under strict complementarity. Preliminary numerical results are reported for DIMACS second-order cone programs, which confirm the favorable theoretical properties of the method.

This paper proposes a new regularization approach, referred to as the Lagrangian regularization approach, which aims to circumvent the nondifferentiability of a positively homogeneous function δ ( · | R - m ) in a conceptual unconstrained reformulation for constrained optimization problems. With the appropriate choices of regularizing function, we obtain a family of smooth functions that include, as special cases, the existing penalty and barrier functions. As such, our approach can be used as an instrumental tool to resolve the nondifferentiability of δ ( · | R - m ) as well as a unified way to construct penalty functions. For convex programming cases, we present its global convergence analysis.

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

The interaction Frohlich-like Hamiltonian between an electron and interface optical phonons in a four-layer heterostructure (FLHS) is obtained by means of the orthonormal eigenmodes of interface optical phonons given recently. The electron-phonon coupling functions and their plots for a FLHS and its special cases, an asymmetric trilayer heterostructure (asymmetric single quantum well) and a step quantum well, are given and discussed. We find that the three branches of higher-frequencies are more important for the electron-phonon interaction than the other three in the six branches of interface modes in a FLHS. Moreover, we also find that the situation in an asymmetric trilayer heterostructure is similar to that of a FLHS.

We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H , and then define a one-parametric class of complementarity functions Φ t on H × H with the parameter t ∈ [ 0 , 2 ) . We show that the squared norm of Φ t with t ∈ ( 0 , 2 ) is a continuously F(rechet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (respectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.