Jein-Shan Chen - Academia.edu (original) (raw)

Papers by Jein-Shan Chen

Nonsymmetric cone program and its corresponding complementarity problem have long been mysterious... more Nonsymmetric cone program and its corresponding complementarity problem have long been mysterious to optimization researchers because of no unified analysis technique to handle these cones. Nonetheless, merit function approach is a popular method to deal with general conic complementarity problem, for which the key lies on constructing appropriate merit functions. In this paper, we focus on a special class of nonsymmetric cone complementarity problem, that is, the ellipsoidal cone complementarity problem (ECCP). We not only show the readers how to construct merit functions for solving the ellipsoidal cone complementarity problem, but also we study the conditions under which the level sets of the corresponding merit functions are bounded. In addition, we assert that these merit functions provide an error bound for the ellipsoidal cone complementarity problem. All these results build up a theoretical basis for the merit function method for solving ellipsoidal cone complementarity prob...

Asia-Pacific Journal of Operational Research, 2007

In this paper, we study several NCP-functions for the nonlinear complementarity problem (NCP) whi... more In this paper, we study several NCP-functions for the nonlinear complementarity problem (NCP) which are indeed based on the generalized Fischer–Burmeister function, ϕp(a, b) = ||(a, b)||p - (a + b). It is well known that the NCP can be reformulated as an equivalent unconstrained minimization by means of merit functions involving NCP-functions. Thus, we aim to investigate some important properties of these NCP-functions that will be used in solving and analyzing the reformulation of the NCP.

Journal of Nonlinear and Convex Analysis, 2019

In this paper, we look into the detailed properties of four discrete-type families of NCP-functio... more In this paper, we look into the detailed properties of four discrete-type families of NCP-functions, which are newly discovered in recent literature. With the discrete-oriented feature, we are motivated to know what differences there are compared to the traditional NCPfunctions. The properties obtained in this paper not only explain the difference but also provide background bricks for designing solution methods based on such discrete-type families of NCP-functions.

IEEE Transactions on Neural Networks and Learning Systems, 2020

We propose an efficient neural network for solving the second-order cone constrained variational ... more We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOC-CVI for short). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality, which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Specifically, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.

In this paper, we revisit the concept of r-convex functions which were studied in 1970s. We prese... more In this paper, we revisit the concept of r-convex functions which were studied in 1970s. We present several novel examples of r-convex functions that are new to the existing literature. In particular, for any given r, we show examples which are r-convex functions. In addition, we extend the concepts of r-convexity and quasi-convexity to the setting associated with second-order cone. Characterizations about such new functions are established. These generalizations will be useful in dealing with optimization problems involved in second-order cones.

arXiv (Cornell University), Jun 6, 2021

A novel approach for solving the general absolute value equation Ax + B|x| = c where A, B ∈ IR m×... more A novel approach for solving the general absolute value equation Ax + B|x| = c where A, B ∈ IR m×n and c ∈ IR m is presented. We reformulate the equation as a feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on A and B. Furthermore, we prove linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with m = n, both theoretically and numerically.

Optimization, 2014

This paper conducts variational analysis of circular programs, which form a new class of optimiza... more This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming important for optimization theory and its applications. First we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/coderivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.

Set-Valued and Variational Analysis, 2016

The circular cone L θ is not self-dual under the standard inner product and includes second-order... more The circular cone L θ is not self-dual under the standard inner product and includes second-order cone as a special case. In this paper, we focus on the monotonicity of f L θ and circular cone monotonicity of f. Their relationship is discussed as well. Our results show that the angle θ plays a different role in these two concepts.

Mathematical Methods of Operations Research, 2006

Recently Tseng [Merit function for semidefinite complementarity, Mathematical Programming, 83, pp... more Recently Tseng [Merit function for semidefinite complementarity, Mathematical Programming, 83, pp. 159-185, 1998] extended a class of merit functions, proposed by Z. Luo and P. Tseng [A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, pp. 204-225, 1997], for the nonlinear complementarity problem (NCP) to the semidefinite complementarity problem (SDCP) and showed several related properties. In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases. In addition, we study another class of merit functions which are based on a slight modification of the aforementioned class of merit functions. Both classes of merit functions provide an error bound for the SOCCP and have bounded level sets.

Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity ... more Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity problem (NCP) and the semidefinite complementarity problem (SDCP) to the second-order cone commplementarity problem (SOCCP) and showed several favorable properties. In this paper, we extend a merit function for the NCP studied by Yamada, Yamashita, and Fukushima to the SOCCP and show that the SOCCP is equivalent to an unconstrained smooth minimization via this new merit function. Furthermore, we study conditions under which the new merit function provides a global error bound which plays an important role in analyzing the convergence rate of iterative methods for solving the SOCCP; and conditions under which the new merit function has bounded level sets which ensures that the sequence generated by a descent method has at least one accumulation point.

Mathematical Programming, 2005

A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as... more A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as the global minimization of a certain merit function over IR n. A popular choice of the merit function is the squared norm of the Fischer-Burmeister function, shown to be smooth over IR n and, for monotone NCP, each stationary point is a solution of the NCP. This merit function and its analysis were subsequently extended to the semidefinite complementarity problem (SDCP), although only differentiability, not continuous differentiability, was established. In this paper, we extend this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP). Although SOCCP is reducible to a SDCP, the reduction does not allow for easy translation of the analysis from SDCP to SOCCP. Instead, our analysis exploits properties of the Jordan product and spectral factorization associated with the second-order cone. We also report preliminary numerical experience with solving DIMACS second-order cone programs using a limited-memory BFGS method to minimize the merit function.

Pacific Journal of Optimization, 2012

This paper makes a survey on SOC complementarity functions and related solution methods for the s... more This paper makes a survey on SOC complementarity functions and related solution methods for the second-order cone programming (SOCP) and second-order cone complementarity problem (SOCCP). Specifically, we discuss the properties of four classes of popular merit functions, and study the theoretical results of associated merit function methods and numerical behaviors in the solution of convex SOCPs. Then, we present suitable nonsinguarity conditions for the B-subdifferentials of the natural residual (NR) and Fischer-Burmeister (FB) nonsmooth system reformulations at a (locally) optimal solution, and test the numerical behavior of a globally convergent FB semismooth Newton method. Finally, we survey the properties of smoothing functions of the NR and FB SOC complementarity functions, and provide numerical comparisons of the smoothing Newton methods based on them. The theoretical results and numerical experience of this paper provide a comprehensive view on the development of this field ...

Journal of Global Optimization

It is known that the analysis to tackle with non-symmetric cone optimization is quite different f... more It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones. However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conicconvexity, conic-monotonicity and etc. In this paper, motivated by Chares Robert's thesis [Chares, R.: Cones and interior-point algorithms for structured convex optimization involving powers and exponentials. PhD thesis, UCL-Universite Catholique de Louvain (2009)], we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart. Keywords Moreau decomposition theorem • power cone • exponential cone • non-symmetric cones. Mathematics Subject Classification (2000) 49M27 • 90C25. 1 Introduction Consider the following two core non-symmetric cones K α := (x 1 ,x) ∈ R × R 2 |x 1 | ≤x α 1 1x α 2 2 ,x 1 ≥ 0,x 2 ≥ 0 , (1) K exp := cl (x 1 ,x) ∈ R × R 2 x 1 ≥x 2 • exp x 1 x 2 ,x 2 > 0, x 1 ≥ 0 , (2)

Neurocomputing, 2016

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with... more This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized Fischer-Burmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

Computational Optimization and Applications, 2015

Operations Research Letters, 2015

ABSTRACT In contrast to the generalized Fischer-Burmeister function that is a natural extension o... more ABSTRACT In contrast to the generalized Fischer-Burmeister function that is a natural extension of the popular Fischer-Burmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized Fischer-Burmeister function.

There are three popular approaches, merit functions approach, nonsmooth functions approach, and s... more There are three popular approaches, merit functions approach, nonsmooth functions approach, and smoothing methods approach, for the second-order cone complementarity problem (SOCCP). In this article, we survey recent results on the most popular approach, merit functions approach. In particular, we investigate and present several merit functions for SOCCP. We also cdot\cdotcdot propose some open questions for future study.

Mathematical Problems in Engineering

This paper focuses on solving the quadratic programming problems with second-order cone constrain... more This paper focuses on solving the quadratic programming problems with second-order cone constraints (SOCQP) and the second-order cone constrained variational inequality (SOCCVI) by using the neural network. More specifically, a neural network model based on two discrete-type families of SOC complementarity functions associated with second-order cone is proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of SOCQP and SOCCVI. The two discrete-type SOC complementarity functions are newly explored. The neural network uses the two discrete-type families of SOC complementarity functions to achieve two unconstrained minimizations which are the merit functions of the Karuch-Kuhn-Tucker equations for SOCQP and SOCCVI. We show that the merit functions for SOCQP and SOCCVI are Lyapunov functions and this neural network is asymptotically stable. The main contribution of this paper lies on its simulation part because we observe a different numerical performance from the existing one. I...

Abstract and Applied Analysis, 2014

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to it... more The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

Abstract and Applied Analysis, 2014

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to it... more The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

Nonsymmetric cone program and its corresponding complementarity problem have long been mysterious... more Nonsymmetric cone program and its corresponding complementarity problem have long been mysterious to optimization researchers because of no unified analysis technique to handle these cones. Nonetheless, merit function approach is a popular method to deal with general conic complementarity problem, for which the key lies on constructing appropriate merit functions. In this paper, we focus on a special class of nonsymmetric cone complementarity problem, that is, the ellipsoidal cone complementarity problem (ECCP). We not only show the readers how to construct merit functions for solving the ellipsoidal cone complementarity problem, but also we study the conditions under which the level sets of the corresponding merit functions are bounded. In addition, we assert that these merit functions provide an error bound for the ellipsoidal cone complementarity problem. All these results build up a theoretical basis for the merit function method for solving ellipsoidal cone complementarity prob...

Asia-Pacific Journal of Operational Research, 2007

In this paper, we study several NCP-functions for the nonlinear complementarity problem (NCP) whi... more In this paper, we study several NCP-functions for the nonlinear complementarity problem (NCP) which are indeed based on the generalized Fischer–Burmeister function, ϕp(a, b) = ||(a, b)||p - (a + b). It is well known that the NCP can be reformulated as an equivalent unconstrained minimization by means of merit functions involving NCP-functions. Thus, we aim to investigate some important properties of these NCP-functions that will be used in solving and analyzing the reformulation of the NCP.

Journal of Nonlinear and Convex Analysis, 2019

In this paper, we look into the detailed properties of four discrete-type families of NCP-functio... more In this paper, we look into the detailed properties of four discrete-type families of NCP-functions, which are newly discovered in recent literature. With the discrete-oriented feature, we are motivated to know what differences there are compared to the traditional NCPfunctions. The properties obtained in this paper not only explain the difference but also provide background bricks for designing solution methods based on such discrete-type families of NCP-functions.

IEEE Transactions on Neural Networks and Learning Systems, 2020

We propose an efficient neural network for solving the second-order cone constrained variational ... more We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOC-CVI for short). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality, which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Specifically, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.

In this paper, we revisit the concept of r-convex functions which were studied in 1970s. We prese... more In this paper, we revisit the concept of r-convex functions which were studied in 1970s. We present several novel examples of r-convex functions that are new to the existing literature. In particular, for any given r, we show examples which are r-convex functions. In addition, we extend the concepts of r-convexity and quasi-convexity to the setting associated with second-order cone. Characterizations about such new functions are established. These generalizations will be useful in dealing with optimization problems involved in second-order cones.

arXiv (Cornell University), Jun 6, 2021

A novel approach for solving the general absolute value equation Ax + B|x| = c where A, B ∈ IR m×... more A novel approach for solving the general absolute value equation Ax + B|x| = c where A, B ∈ IR m×n and c ∈ IR m is presented. We reformulate the equation as a feasibility problem which we solve via the method of alternating projections (MAP). The fixed points set of the alternating projections map is characterized under nondegeneracy conditions on A and B. Furthermore, we prove linear convergence of the algorithm. Unlike most of the existing approaches in the literature, the algorithm presented here is capable of handling problems with m = n, both theoretically and numerically.

Optimization, 2014

This paper conducts variational analysis of circular programs, which form a new class of optimiza... more This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming important for optimization theory and its applications. First we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/coderivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.

Set-Valued and Variational Analysis, 2016

The circular cone L θ is not self-dual under the standard inner product and includes second-order... more The circular cone L θ is not self-dual under the standard inner product and includes second-order cone as a special case. In this paper, we focus on the monotonicity of f L θ and circular cone monotonicity of f. Their relationship is discussed as well. Our results show that the angle θ plays a different role in these two concepts.

Mathematical Methods of Operations Research, 2006

Recently Tseng [Merit function for semidefinite complementarity, Mathematical Programming, 83, pp... more Recently Tseng [Merit function for semidefinite complementarity, Mathematical Programming, 83, pp. 159-185, 1998] extended a class of merit functions, proposed by Z. Luo and P. Tseng [A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, pp. 204-225, 1997], for the nonlinear complementarity problem (NCP) to the semidefinite complementarity problem (SDCP) and showed several related properties. In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases. In addition, we study another class of merit functions which are based on a slight modification of the aforementioned class of merit functions. Both classes of merit functions provide an error bound for the SOCCP and have bounded level sets.

Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity ... more Recently, J.-S. Chen and P. Tseng extended two merit functions for the nonlinear complementarity problem (NCP) and the semidefinite complementarity problem (SDCP) to the second-order cone commplementarity problem (SOCCP) and showed several favorable properties. In this paper, we extend a merit function for the NCP studied by Yamada, Yamashita, and Fukushima to the SOCCP and show that the SOCCP is equivalent to an unconstrained smooth minimization via this new merit function. Furthermore, we study conditions under which the new merit function provides a global error bound which plays an important role in analyzing the convergence rate of iterative methods for solving the SOCCP; and conditions under which the new merit function has bounded level sets which ensures that the sequence generated by a descent method has at least one accumulation point.

Mathematical Programming, 2005

A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as... more A popular approach to solving the nonlinear complementarity problem (NCP) is to reformulate it as the global minimization of a certain merit function over IR n. A popular choice of the merit function is the squared norm of the Fischer-Burmeister function, shown to be smooth over IR n and, for monotone NCP, each stationary point is a solution of the NCP. This merit function and its analysis were subsequently extended to the semidefinite complementarity problem (SDCP), although only differentiability, not continuous differentiability, was established. In this paper, we extend this merit function and its analysis, including continuous differentiability, to the second-order cone complementarity problem (SOCCP). Although SOCCP is reducible to a SDCP, the reduction does not allow for easy translation of the analysis from SDCP to SOCCP. Instead, our analysis exploits properties of the Jordan product and spectral factorization associated with the second-order cone. We also report preliminary numerical experience with solving DIMACS second-order cone programs using a limited-memory BFGS method to minimize the merit function.

Pacific Journal of Optimization, 2012

This paper makes a survey on SOC complementarity functions and related solution methods for the s... more This paper makes a survey on SOC complementarity functions and related solution methods for the second-order cone programming (SOCP) and second-order cone complementarity problem (SOCCP). Specifically, we discuss the properties of four classes of popular merit functions, and study the theoretical results of associated merit function methods and numerical behaviors in the solution of convex SOCPs. Then, we present suitable nonsinguarity conditions for the B-subdifferentials of the natural residual (NR) and Fischer-Burmeister (FB) nonsmooth system reformulations at a (locally) optimal solution, and test the numerical behavior of a globally convergent FB semismooth Newton method. Finally, we survey the properties of smoothing functions of the NR and FB SOC complementarity functions, and provide numerical comparisons of the smoothing Newton methods based on them. The theoretical results and numerical experience of this paper provide a comprehensive view on the development of this field ...

Journal of Global Optimization

It is known that the analysis to tackle with non-symmetric cone optimization is quite different f... more It is known that the analysis to tackle with non-symmetric cone optimization is quite different from the way to deal with symmetric cone optimization due to the discrepancy between these types of cones. However, there are still common concepts for both optimization problems, for example, the decomposition with respect to the given cone, smooth and nonsmooth analysis for the associated conic function, conicconvexity, conic-monotonicity and etc. In this paper, motivated by Chares Robert's thesis [Chares, R.: Cones and interior-point algorithms for structured convex optimization involving powers and exponentials. PhD thesis, UCL-Universite Catholique de Louvain (2009)], we consider the decomposition issue of two core non-symmetric cones, in which two types of decomposition formulae will be proposed, one is adapted from the well-known Moreau decomposition theorem and the other follows from geometry properties of the given cones. As a byproduct, we also establish the conic functions of these cones and generalize the power cone case to its high-dimensional counterpart. Keywords Moreau decomposition theorem • power cone • exponential cone • non-symmetric cones. Mathematics Subject Classification (2000) 49M27 • 90C25. 1 Introduction Consider the following two core non-symmetric cones K α := (x 1 ,x) ∈ R × R 2 |x 1 | ≤x α 1 1x α 2 2 ,x 1 ≥ 0,x 2 ≥ 0 , (1) K exp := cl (x 1 ,x) ∈ R × R 2 x 1 ≥x 2 • exp x 1 x 2 ,x 2 > 0, x 1 ≥ 0 , (2)

Neurocomputing, 2016

This paper proposes a neural network approach to efficiently solve nonlinear convex programs with... more This paper proposes a neural network approach to efficiently solve nonlinear convex programs with the second-order cone constraints. The neural network model is designed by the generalized Fischer-Burmeister function associated with second-order cone. We study the existence and convergence of the trajectory for the considered neural network. Moreover, we also show stability properties for the considered neural network, including the Lyapunov stability, the asymptotic stability and the exponential stability. Illustrative examples give a further demonstration for the effectiveness of the proposed neural network. Numerical performance based on the parameter being perturbed and numerical comparison with other neural network models are also provided. In overall, our model performs better than two comparative methods.

Computational Optimization and Applications, 2015

Operations Research Letters, 2015

ABSTRACT In contrast to the generalized Fischer-Burmeister function that is a natural extension o... more ABSTRACT In contrast to the generalized Fischer-Burmeister function that is a natural extension of the popular Fischer-Burmeister function NCP-function, the generalized natural residual NCP-function based on discrete extension, recently proposed by Chen, Ko, and Wu, does not possess symmetric graph. In this paper we symmetrize the generalized natural residual NCP-function, and construct not only new NCP-functions and merit functions for the nonlinear complementarity problem, but also provide parallel functions to the generalized Fischer-Burmeister function.

There are three popular approaches, merit functions approach, nonsmooth functions approach, and s... more There are three popular approaches, merit functions approach, nonsmooth functions approach, and smoothing methods approach, for the second-order cone complementarity problem (SOCCP). In this article, we survey recent results on the most popular approach, merit functions approach. In particular, we investigate and present several merit functions for SOCCP. We also cdot\cdotcdot propose some open questions for future study.

Mathematical Problems in Engineering

This paper focuses on solving the quadratic programming problems with second-order cone constrain... more This paper focuses on solving the quadratic programming problems with second-order cone constraints (SOCQP) and the second-order cone constrained variational inequality (SOCCVI) by using the neural network. More specifically, a neural network model based on two discrete-type families of SOC complementarity functions associated with second-order cone is proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of SOCQP and SOCCVI. The two discrete-type SOC complementarity functions are newly explored. The neural network uses the two discrete-type families of SOC complementarity functions to achieve two unconstrained minimizations which are the merit functions of the Karuch-Kuhn-Tucker equations for SOCQP and SOCCVI. We show that the merit functions for SOCQP and SOCCVI are Lyapunov functions and this neural network is asymptotically stable. The main contribution of this paper lies on its simulation part because we observe a different numerical performance from the existing one. I...

Abstract and Applied Analysis, 2014

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to it... more The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

Abstract and Applied Analysis, 2014

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to it... more The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. LetLθdenote the circular cone inRn. For a functionffromRtoR, one can define a corresponding vector-valued functionfLθonRnby applyingfto the spectral values of the spectral decomposition ofx∈Rnwith respect toLθ. In this paper, we study properties that this vector-valued function inherits fromf, including Hölder continuity,B-subdifferentiability,ρ-order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.