In this paper, we have introduced the concepts of pseudomonotonicity properties for nonlinear transformations defined on Euclidean Jordan algebras. The implications between this property and other P-properties have be...In this paper, we have introduced the concepts of pseudomonotonicity properties for nonlinear transformations defined on Euclidean Jordan algebras. The implications between this property and other P-properties have been studied. More importantly, we have solved the solvability problem of the nonlinear pseudomonotone complementarity problems over symmetric cones.展开更多
The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity proble...The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.展开更多
In this paper, we introduce a new class of two-parametric penalized function,which includes the penalized minimum function and the penalized Fischer-Burmeister function over symmetric cone complementarity problems. We...In this paper, we introduce a new class of two-parametric penalized function,which includes the penalized minimum function and the penalized Fischer-Burmeister function over symmetric cone complementarity problems. We propose that this class of function is a class of complementarity functions(C-function). Moreover, its merit function has bounded level set under a weak condition.展开更多
We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the saf...We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al[18]. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε-1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.展开更多
In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N( t1, t2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of t...In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N( t1, t2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r1.5 log ε-1) for the Nesterov-Todd (NT) direction, and O(r2 log ε-1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε 〉 0 is the required precision. If starting with a feasible point (x0, y0, s0) in N(t1, t2, η), the complexity bound is O( √ r log ε-1) for the NT direction, and O(r log ε-1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.展开更多
In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program ((SCLP) for short) by making use of an augmented system of its optimality conditions. The algorithm only needs...In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program ((SCLP) for short) by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of (SCLP). Furthermore, if (SCLP) has a solution, then the algorithm will generate a solution of (SCLP), and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of (SCLP).展开更多
In this paper,we present an infeasible-interior-point algorithm,based on a new wide neighborhood for symmetric cone programming.We treat the classical Newton direction as the sum of two other directions,and equip them...In this paper,we present an infeasible-interior-point algorithm,based on a new wide neighborhood for symmetric cone programming.We treat the classical Newton direction as the sum of two other directions,and equip them with different step sizes.We prove the complexity bound of the new algorithm for the Nesterov-Todd(NT)direction,and the xs and sx directions.The complexity bounds obtained here are the same as small neighborhood infeasible-interior-point algorithms over symmetric cones.展开更多
This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming(SCP)based on wide neighborhoods and new directions with a parameterθ.When the parameterθ=1,the direc...This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming(SCP)based on wide neighborhoods and new directions with a parameterθ.When the parameterθ=1,the direction is exactly the classical Newton direction.When the parameterθis independent of the rank of the associated Euclidean Jordan algebra,the algorithm terminates in at most O(κr logε−1)iterations,which coincides with the best known iteration bound for the classical wide neighborhood algorithms.When the parameterθ=√n/βτand Nesterov–Todd search direction is used,the algorithm has O(√r logε−1)iteration complexity,the best iteration complexity obtained so far by any interior-point method for solving SCP.To our knowledge,this is the first time that a class of interior-point algorithms including the classical wide neighborhood path-following algorithm is proposed and analyzed over symmetric cone.展开更多
The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems as- serts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA^* is posi...The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems as- serts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a E V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.展开更多
基金supported by the Natural Science Basic Research Program of Shaanxi (Program No. 2023-JCYB-048)the National Natural Science Foundation of China (Program No. 11601406)。
文摘In this paper, we have introduced the concepts of pseudomonotonicity properties for nonlinear transformations defined on Euclidean Jordan algebras. The implications between this property and other P-properties have been studied. More importantly, we have solved the solvability problem of the nonlinear pseudomonotone complementarity problems over symmetric cones.
基金Supported by the Funds of Ministry of Education of China for PhD (20020141013)the NNSF of China (10471015).
文摘The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.
文摘In this paper, we introduce a new class of two-parametric penalized function,which includes the penalized minimum function and the penalized Fischer-Burmeister function over symmetric cone complementarity problems. We propose that this class of function is a class of complementarity functions(C-function). Moreover, its merit function has bounded level set under a weak condition.
基金Supported by the National Natural Science Foundation of China(11471102,61301229)Supported by the Natural Science Foundation of Henan University of Science and Technology(2014QN039)
文摘We establish polynomial complexity bounds of the Mehrotra-type predictorcorrector algorithms for linear programming over symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al[18]. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε-1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.
基金Supported by the National Natural Science Foundation of China(No.11471102)the Key Basic Research Foundation of the Higher Education Institutions of Henan Province(No.16A110012)
文摘In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N( t1, t2, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r1.5 log ε-1) for the Nesterov-Todd (NT) direction, and O(r2 log ε-1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε 〉 0 is the required precision. If starting with a feasible point (x0, y0, s0) in N(t1, t2, η), the complexity bound is O( √ r log ε-1) for the NT direction, and O(r log ε-1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.
基金Supported by the National Natural Science Foundation of China(No.11171252,11301375 and 71301118)Research Fund for the Doctoral Program of Higher Education of China(No.20120032120076)Tianjin Planning Program of Philosophy and Social Science(No.TJTJ11-004)
文摘In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program ((SCLP) for short) by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of (SCLP). Furthermore, if (SCLP) has a solution, then the algorithm will generate a solution of (SCLP), and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of (SCLP).
基金the National Natural Science Foundation of China(Nos.11471102,11426091,and 61179040)the Natural Science Foundation of Henan University of Science and Technology(No.2014QN039)Key Basic Research Foundation of the Higher Education Institutions of Henan Province(No.16A110012).
文摘In this paper,we present an infeasible-interior-point algorithm,based on a new wide neighborhood for symmetric cone programming.We treat the classical Newton direction as the sum of two other directions,and equip them with different step sizes.We prove the complexity bound of the new algorithm for the Nesterov-Todd(NT)direction,and the xs and sx directions.The complexity bounds obtained here are the same as small neighborhood infeasible-interior-point algorithms over symmetric cones.
基金the National Natural Science Foundation of China(No.11471102)the Key Basic Research Foundation of the Higher Education Institutions of Henan Province(No.16A110012)。
文摘This paper presents a class of primal-dual path-following interior-point algorithms for symmetric cone programming(SCP)based on wide neighborhoods and new directions with a parameterθ.When the parameterθ=1,the direction is exactly the classical Newton direction.When the parameterθis independent of the rank of the associated Euclidean Jordan algebra,the algorithm terminates in at most O(κr logε−1)iterations,which coincides with the best known iteration bound for the classical wide neighborhood algorithms.When the parameterθ=√n/βτand Nesterov–Todd search direction is used,the algorithm has O(√r logε−1)iteration complexity,the best iteration complexity obtained so far by any interior-point method for solving SCP.To our knowledge,this is the first time that a class of interior-point algorithms including the classical wide neighborhood path-following algorithm is proposed and analyzed over symmetric cone.
基金Supported partially by National Natural Science Foundation of China (Grant Nos. 10871056 and 10971150)Science Research Foundation in Harbin Institute of Technology (Grant No. HITC200708)
文摘The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems as- serts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V. Given any element a E V, we consider the corresponding Lyapunov transformation La and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R0-property for La and show that La has the R0-property if and only if a is invertible. Finally, we provide La with some characterizations of the E0-property and the nondegeneracy property.