Jinchuan Zhou

Circular cone convexity and some inequalities associated with circular cones

The study of this paper consists of two aspects. One is characterizing the so-called circular cone convexity of f by exploiting the second-order differentiability of fLθ; the other is introducing the concepts of determinant and trace associated with circular cone and establishing their basic inequalities. These results show the essential role played by the angle θ, which gives us a new insight when looking into properties about circular cone.MSC:26A27, 26B05, 26B35, 49J52, 90C33, 65K05.

Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations

In this paper, some new nonlinear delay integral inequalities on time scales are established, which provide a handy tool in the research of boundedness of unknown functions in delay dynamic equations on time scales. The established results generalize some of the results in Lipovan [J. Math. Anal. Appl. 322, 349-358 (2006)], Pachpatte [J. Math. Anal. Appl. 251, 736-751 (2000)], Li [Comput. Math. Appl. 59, 1929-1936 (2010)], and Sun [J. Math. Anal. Appl. 301, 265-275 (2005)].MSC 2010: 26E70; 26D15; 26D10.

A new class of exact penalty functions and penalty algorithms

For nonlinear programming problems, we propose a new class of smooth exact penalty functions, which includes both barrier-type and exterior-type penalty functions as special cases. We develop necessary and sufficient conditions for exact penalty property and inverse proposition of exact penalization, respectively. Furthermore, we establish the equivalent relationship between these penalty functions and classical simple exact penalty functions in the sense of exactness property. In addition, a feasible penalty function algorithm is proposed. The convergence analysis of the algorithm is presented, including the global convergence property and finite termination property. Finally, numerical results are reported.

Variational analysis of circular cone programs

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/co-derivatives 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.

A smoothing-type algorithm for the second-order cone complementarity problem with a new nonmonotone line search

The second-order cone complementarity problem (denoted by SOCCP) can be effectively solved by smoothing-type algorithms, which in general are designed based on some monotone line search. In this paper, based on a new smoothing function of the Fischer–Burmeister function, we propose a smoothing-type algorithm for solving the SOCCP. The proposed algorithm uses a new nonmonotone line search scheme, which contains the usual monotone line search as a special case. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported which indicate the effectiveness of the proposed algorithm.

The Vector-Valued Functions Associated with Circular Cones

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. Let denote the circular cone in . For a function from to , one can define a corresponding vector-valued function on by applying to the spectral values of the spectral decomposition of with respect to . In this paper, we study properties that this vector-valued function inherits from , including Holder continuity, -subdifferentiability, -order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

Saddle point and exact penalty representation for generalized proximal Lagrangians

In this paper, we introduce a generalized proximal Lagrangian function for the constrained nonlinear programming problem and discuss existence of its saddle points. In particular, the local saddle point is obtained by using the second-order sufficient conditions, and the global saddle point is given without requiring compactness of constraint set and uniqueness of the optimal solution. Finally, we establish equivalent relationship between global saddle points and exact penalty representations.

On the existence of saddle points for nonlinear second-order cone programming problems

In this paper, we study the existence of local and global saddle points for nonlinear second-order cone programming problems. The existence of local saddle points is developed by using the second-order sufficient conditions, in which a sigma-term is added to reflect the curvature of second-order cone. Furthermore, by dealing with the perturbation of the primal problem, we establish the existence of global saddle points, which can be applicable for the case of multiple optimal solutions. The close relationship between global saddle points and exact penalty representations are discussed as well.

Some Delay Integral Inequalities on Time Scales and Their Applications in the Theory of Dynamic Equations

We establish some delay integral inequalities on time scales, which on one hand provide a handy tool in the study of qualitative as well as quantitative properties of solutions of certain delay dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.

On Set-Valued Complementarity Problems

This paper investigates the set-valued complementarity problems (SVCP) which poses rather different features from those that classical complementarity problems hold, due to tthe fact that he index set is not fixed, but dependent on . While comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP. Moreover, properties of merit functions for SVCP are studied, such being as level bounded and error bounded. Finally, some possible research directions are discussed.