Luo, ZhiQuan
Overview
Works:  7 works in 22 publications in 1 language and 35 library holdings 

Roles:  Author 
Publication Timeline
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Most widely held works by
ZhiQuan Luo
On the extensions of FrankWolfe theorem by
ZhiQuan Luo(
Book
)
5 editions published in 1997 in English and held by 9 WorldCat member libraries worldwide
In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the socalled FrankWolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasiconvex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty
5 editions published in 1997 in English and held by 9 WorldCat member libraries worldwide
In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the socalled FrankWolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasiconvex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty
Superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming by
ZhiQuan Luo(
Book
)
4 editions published in 1996 in English and held by 7 WorldCat member libraries worldwide
This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends to zero. The interior point algorithm considered here closely resembles the MizunoToddYe predictorcorrector method for linear programming which is known to be quadratically convergent. It is shown that when the iterates are well centered, the duality gap is reduced superlinearly after each predictor step. Indeed, if each predictor step is succeeded by $r$ consecutive corrector steps then the predictor reduces the duality gap superlinearly with order $\frac{2}{1+2^{2r}}$. The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primaldual central path converges to the analytic center of the primaldual optimal solution set, and the distance from any point on the central path to this analytic center is bounded by the duality gap
4 editions published in 1996 in English and held by 7 WorldCat member libraries worldwide
This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends to zero. The interior point algorithm considered here closely resembles the MizunoToddYe predictorcorrector method for linear programming which is known to be quadratically convergent. It is shown that when the iterates are well centered, the duality gap is reduced superlinearly after each predictor step. Indeed, if each predictor step is succeeded by $r$ consecutive corrector steps then the predictor reduces the duality gap superlinearly with order $\frac{2}{1+2^{2r}}$. The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primaldual central path converges to the analytic center of the primaldual optimal solution set, and the distance from any point on the central path to this analytic center is bounded by the duality gap
Duality and selfduality for conic convex programming by
ZhiQuan Luo(
Book
)
4 editions published in 1996 in English and held by 6 WorldCat member libraries worldwide
This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, these objective values in fact converge to the optimal value of the original problem. When the problem is neither strongly infeasible nor endowed with a complementary pair, we completely specify the asymptotic behavior of an indicator in relation to the status of the original problem, namely whether the problem (1) is weakly infeasible, (2) is feasible but with a positive duality gap, (3) has no duality gap nor complementary solution pair
4 editions published in 1996 in English and held by 6 WorldCat member libraries worldwide
This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, these objective values in fact converge to the optimal value of the original problem. When the problem is neither strongly infeasible nor endowed with a complementary pair, we completely specify the asymptotic behavior of an indicator in relation to the status of the original problem, namely whether the problem (1) is weakly infeasible, (2) is feasible but with a positive duality gap, (3) has no duality gap nor complementary solution pair
Complexity analysis of a logarithmic barrier decomposition method for semiinfinite linear programming by
ZhiQuan Luo(
Book
)
2 editions published in 1997 in English and held by 4 WorldCat member libraries worldwide
2 editions published in 1997 in English and held by 4 WorldCat member libraries worldwide
Duality results for conic convex programming by
ZhiQuan Luo(
Book
)
2 editions published in 1997 in English and held by 3 WorldCat member libraries worldwide
This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone in finite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include GordonStiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed
2 editions published in 1997 in English and held by 3 WorldCat member libraries worldwide
This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone in finite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include GordonStiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed
Conic convex programming and selfdual embedding by
Z.Q Luo(
Book
)
2 editions published in 1998 in English and held by 2 WorldCat member libraries worldwide
How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big $M$ technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the selfdual embedding technique proposed by Ye, Todd and Mizuno. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded selfdual problem. The embedded selfdual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the socalled strict complementarity property, it causes difficulties in identifying solutions for the original problem, based on solutions for the embedded selfdual system. We provide numerous examples from semidefinite programming to illustrate various possibilities which have no analogue in the linear programming case
2 editions published in 1998 in English and held by 2 WorldCat member libraries worldwide
How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big $M$ technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the selfdual embedding technique proposed by Ye, Todd and Mizuno. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded selfdual problem. The embedded selfdual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the socalled strict complementarity property, it causes difficulties in identifying solutions for the original problem, based on solutions for the embedded selfdual system. We provide numerous examples from semidefinite programming to illustrate various possibilities which have no analogue in the linear programming case
Multivariate nonnegative quadratic mapping by
ZhiQuan Luo(
Book
)
2 editions published in 2003 in English and held by 1 WorldCat member library worldwide
2 editions published in 2003 in English and held by 1 WorldCat member library worldwide
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