In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transform...In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transformation to derive the search direction.It is shown that the proximity measure reduces quadratically at each iteration.Moreover,the iteration bound of the algorithm is as good as the best-known polynomial complexity for these types of problems.Furthermore,numerical results are presented to show the efficiency of the proposed algorithm.展开更多
In this paper,a fu-Newton step interior-point algorithm is proposed for solving P_(*)(k)-linear complementarity problem based on a new search direction,which is an extension of Grimes'algorithm.It is proved that t...In this paper,a fu-Newton step interior-point algorithm is proposed for solving P_(*)(k)-linear complementarity problem based on a new search direction,which is an extension of Grimes'algorithm.It is proved that the number of iterations of the algorithm is O(n^(1/2)(1+4κ)logn/ε),which matches the best known iteration bound of the interior-point method for P_(*)(k)-linear complementarity problem.Some numerical results have proved the feasibility and efficiency of the proposed algorithm.展开更多
In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear ...In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear growth term to derive the search direction,and by introducing new technical results and selecting suitable parameters,we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods.Furthermore,numerical results illustrate the efficiency of the proposed method.展开更多
This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one c...This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.展开更多
Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with si...Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.展开更多
In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functio...In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.展开更多
In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barr...In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barrier term. Iteration bounds both for large-and small-update methods are derived, namely, O（nlog（n/c）） and O（√nlog（n/ε））. This new kernel function has simple algebraic expression and the proximity function has not been used before. Analogous to the classical logarithmic kernel function, our complexity analysis is easier than the other pri- mal-dual interior-point methods based on logarithmic barrier functions and recent kernel functions.展开更多
In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the ent...In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O（nlog（（ x^0）~TS^0/ε）） which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.展开更多
In this paper, a primal-dual path-following interior-point algorithm for linearly constrained convex optimization(LCCO) is presented.The algorithm is based on a new technique for finding a class of search directions a...In this paper, a primal-dual path-following interior-point algorithm for linearly constrained convex optimization(LCCO) is presented.The algorithm is based on a new technique for finding a class of search directions and the strategy of the central path.At each iteration, only full-Newton steps are used.Finally, the favorable polynomial complexity bound for the algorithm with the small-update method is deserved, namely, O(√n log n /ε).展开更多
A new iterative method,which is called positive interior-point algorithm,is presented for solving the nonlinear complementarity problems.This method is of the desirable feature of robustness.And the convergence theore...A new iterative method,which is called positive interior-point algorithm,is presented for solving the nonlinear complementarity problems.This method is of the desirable feature of robustness.And the convergence theorems of the algorithm is established.In addition,some numerical results are reported.展开更多
In this paper, a new primal-dual interior-point algorithm for convex quadratic optimization （CQO） based on a kernel function is presented. The proposed function has some properties that are easy for checking. These ...In this paper, a new primal-dual interior-point algorithm for convex quadratic optimization （CQO） based on a kernel function is presented. The proposed function has some properties that are easy for checking. These properties enable us to improve the polynomial complexity bound of a large-update interior-point method （IPM） to O（√n log nlog n/e）, which is the currently best known polynomial complexity bound for the algorithm with the large-update method. Numerical tests were conducted to investigate the behavior of the algorithm with different parameters p, q and θ, where p is the growth degree parameter, q is the barrier degree of the kernel function and θ is the barrier update parameter.展开更多
In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefmite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to...In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefmite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to its growth term increasing linearly. Some new analysis tools were developed which can be used to deal with complexity ＂analysis of the algorithms which use analogous strategy in [5] to design the search directions for the Newton system. The complexity bounds for the algorithms with large- and small-update methodswere obtained, namely,O（qn^（p＋q/q（P＋1）log n/ε and O（q^2√n）log n/ε,respectlvely.展开更多
Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In ...Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In the p aper a predictor-corrector interior-point algorithm for linearly constrained c onvex programming under the predictor-corrector motivation was proposed. In eac h iteration, the algorithm first performs a predictor-step to reduce the dualit y gap and then a corrector-step to keep the points close to the central traject ory. Computations in the algorithm only require that the initial iterate be nonn egative while feasibility or strict feasibility is not required. It is proved th at the algorithm is equivalent to a level-1 perturbed composite Newton method. Numerical experiments on twenty-six standard test problems are made. The result s show that the proposed algorithm is stable and robust.展开更多
The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off betwee...The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.展开更多
Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work t...Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work the authors give a predictor corrector interior point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level 1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.展开更多
In this paper, a corrector-predictor interior-point algorithm is proposed for sym- metric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-p...In this paper, a corrector-predictor interior-point algorithm is proposed for sym- metric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-path step by step and generates a sequence of iter- ates in a wide neighborhood of the central-path. Using the machinery of Euclidean Jordan algebra and the commutative class of search directions, the convergence analysis of the algo- rithm is shown and it is proved that the algorithm has the complexity bound O （√τL） for the well-known Nesterov-Todd search direction and O （τL） for the xs and sx search directions.展开更多
This paper proposes a new full Nesterov-Todd(NT) step infeasible interior-point algorithm for semidefinite programming. Our algorithm uses a specific kernel function, which is adopted by Liu and Sun, to deduce the fea...This paper proposes a new full Nesterov-Todd(NT) step infeasible interior-point algorithm for semidefinite programming. Our algorithm uses a specific kernel function, which is adopted by Liu and Sun, to deduce the feasibility step. By using the step, it is remarkable that in each iteration of the algorithm it needs only one full-NT step, and can obtain an iterate approximate to the central path. Moreover, it is proved that the iterative bound corresponds with the known optimal one for semidefinite optimization problems.展开更多
The choice of self-concordant functions is the key to efficient algorithms for linear and quadratic convex optimizations, which provide a method with polynomial-time iterations to solve linear and quadratic convex opt...The choice of self-concordant functions is the key to efficient algorithms for linear and quadratic convex optimizations, which provide a method with polynomial-time iterations to solve linear and quadratic convex optimization problems. The parameters of a self-concordant barrier function can be used to compute the complexity bound of the proposed algorithm. In this paper, it is proved that the finite barrier function is a local self-concordant barrier function. By deriving the local values of parameters of this barrier function, the desired complexity bound of an interior-point algorithm based on this local self-concordant function for linear optimization problem is obtained. The bound matches the best known bound for small-update methods.展开更多
In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps....In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. We used a specific kernel function to induce the feasibility step. The analysis is more simplified. The iteration bound coincides with the currently best known bound for infeasible interior-point methods.展开更多
A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Un...A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Under the mild conditions for the barrier term, the complexity bound of algorithm in terms of such kernel function and its derivatives is obtained. The approach is actually an extension of the existing work which only used the specific kernel functions for the MLCP.展开更多
基金Supported by the Optimisation Theory and Algorithm Research Team(Grant No.23kytdzd004)University Science Research Project of Anhui Province(Grant No.2024AH050631)the General Programs for Young Teacher Cultivation of Educational Commission of Anhui Province(Grant No.YQYB2023090).
文摘In this paper,we propose a new full-Newton step feasible interior-point algorithm for the special weighted linear complementarity problems.The proposed algorithm employs the technique of algebraic equivalent transformation to derive the search direction.It is shown that the proximity measure reduces quadratically at each iteration.Moreover,the iteration bound of the algorithm is as good as the best-known polynomial complexity for these types of problems.Furthermore,numerical results are presented to show the efficiency of the proposed algorithm.
基金Supported by the Optimization Theory and Algorithm Research Team(23kytdzd004)the General Programs for Young Teacher Cultivation of Educational Commission of Anhui Province of China(YQYB2023090)the University Science Research Project of Anhui Province(2024AH050631)。
文摘In this paper,a fu-Newton step interior-point algorithm is proposed for solving P_(*)(k)-linear complementarity problem based on a new search direction,which is an extension of Grimes'algorithm.It is proved that the number of iterations of the algorithm is O(n^(1/2)(1+4κ)logn/ε),which matches the best known iteration bound of the interior-point method for P_(*)(k)-linear complementarity problem.Some numerical results have proved the feasibility and efficiency of the proposed algorithm.
基金Supported by University Science Research Project of Anhui Province(2023AH052921)Outstanding Youth Talent Project of Anhui Province(gxyq2021254)。
文摘In this paper,a new full-Newton step primal-dual interior-point algorithm for solving the special weighted linear complementarity problem is designed and analyzed.The algorithm employs a kernel function with a linear growth term to derive the search direction,and by introducing new technical results and selecting suitable parameters,we prove that the iteration bound of the algorithm is as good as best-known polynomial complexity of interior-point methods.Furthermore,numerical results illustrate the efficiency of the proposed method.
基金Project supported by the National Science Foundation of China (60574071) the Foundation for University Key Teacher by the Ministry of Education.
文摘This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.
基金Project supported by the National Natural Science Foundation of China （Grant No. 10117733）, the Shanghai Leading Academic Discipline Project （Grant No.J50101）, and the Foundation of Scientific Research for Selecting and Cultivating Young Excellent University Teachers in Shanghai （Grant No.06XPYQ52）
文摘Interior-point methods （IPMs） for linear optimization （LO） and semidefinite optimization （SDO） have become a hot area in mathematical programming in the last decades. In this paper, a new kernel function with simple algebraic expression is proposed. Based on this kernel function, a primal-dual interior-point methods （IPMs） for semidefinite optimization （SDO） is designed. And the iteration complexity of the algorithm as O（n^3/4 log n/ε） with large-updates is established. The resulting bound is better than the classical kernel function, with its iteration complexity O（n log n/ε） in large-updates case.
基金Project supported by Dutch Organization for Scientific Research(Grant No .613 .000 .010)
文摘In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.
基金Supported by the Natural Science Foundation of Hubei Province (2008CDZD47)
文摘In this paper, we design a primal-dual interior-point algorithm for linear optimization. Search directions and proximity function are proposed based on a new kernel function which includes neither growth term nor barrier term. Iteration bounds both for large-and small-update methods are derived, namely, O（nlog（n/c）） and O（√nlog（n/ε））. This new kernel function has simple algebraic expression and the proximity function has not been used before. Analogous to the classical logarithmic kernel function, our complexity analysis is easier than the other pri- mal-dual interior-point methods based on logarithmic barrier functions and recent kernel functions.
基金Supported by the National Natural Science Foundation of China(71471102)
文摘In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O（nlog（（ x^0）~TS^0/ε）） which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.
基金supported by the Shanghai Pujiang Program (Grant No.06PJ14039)the Science Foundation of Shanghai Municipal Commission of Education (Grant No.06NS031)
文摘In this paper, a primal-dual path-following interior-point algorithm for linearly constrained convex optimization(LCCO) is presented.The algorithm is based on a new technique for finding a class of search directions and the strategy of the central path.At each iteration, only full-Newton steps are used.Finally, the favorable polynomial complexity bound for the algorithm with the small-update method is deserved, namely, O(√n log n /ε).
文摘A new iterative method,which is called positive interior-point algorithm,is presented for solving the nonlinear complementarity problems.This method is of the desirable feature of robustness.And the convergence theorems of the algorithm is established.In addition,some numerical results are reported.
基金the Foundation of Scientific Research for Selecting and Cultivating Young Excellent University Teachers in Shanghai (Grant No.06XPYQ52)the Shanghai Pujiang Program (Grant No.06PJ14039)
文摘In this paper, a new primal-dual interior-point algorithm for convex quadratic optimization （CQO） based on a kernel function is presented. The proposed function has some properties that are easy for checking. These properties enable us to improve the polynomial complexity bound of a large-update interior-point method （IPM） to O（√n log nlog n/e）, which is the currently best known polynomial complexity bound for the algorithm with the large-update method. Numerical tests were conducted to investigate the behavior of the algorithm with different parameters p, q and θ, where p is the growth degree parameter, q is the barrier degree of the kernel function and θ is the barrier update parameter.
文摘In the present paper we present a class of polynomial primal-dual interior-point algorithms for semidefmite optimization based on a kernel function. This kernel function is not a so-called self-regular function due to its growth term increasing linearly. Some new analysis tools were developed which can be used to deal with complexity ＂analysis of the algorithms which use analogous strategy in [5] to design the search directions for the Newton system. The complexity bounds for the algorithms with large- and small-update methodswere obtained, namely,O（qn^（p＋q/q（P＋1）log n/ε and O（q^2√n）log n/ε,respectlvely.
文摘Active set method and gradient projection method are curre nt ly the main approaches for linearly constrained convex programming. Interior-po int method is one of the most effective choices for linear programming. In the p aper a predictor-corrector interior-point algorithm for linearly constrained c onvex programming under the predictor-corrector motivation was proposed. In eac h iteration, the algorithm first performs a predictor-step to reduce the dualit y gap and then a corrector-step to keep the points close to the central traject ory. Computations in the algorithm only require that the initial iterate be nonn egative while feasibility or strict feasibility is not required. It is proved th at the algorithm is equivalent to a level-1 perturbed composite Newton method. Numerical experiments on twenty-six standard test problems are made. The result s show that the proposed algorithm is stable and robust.
文摘The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.
文摘Mehrotra's recent suggestion of a predictor corrector variant of primal dual interior point method for linear programming is currently the interior point method of choice for linear programming. In this work the authors give a predictor corrector interior point algorithm for monotone variational inequality problems. The algorithm was proved to be equivalent to a level 1 perturbed composite Newton method. Computations in the algorithm do not require the initial iteration to be feasible. Numerical results of experiments are presented.
基金Shahrekord University for financial supportpartially supported by the Center of Excellence for Mathematics, University of Shahrekord, Shahrekord, Iran
文摘In this paper, a corrector-predictor interior-point algorithm is proposed for sym- metric optimization. The algorithm approximates the central path by an ellipse, follows the ellipsoidal approximation of the central-path step by step and generates a sequence of iter- ates in a wide neighborhood of the central-path. Using the machinery of Euclidean Jordan algebra and the commutative class of search directions, the convergence analysis of the algo- rithm is shown and it is proved that the algorithm has the complexity bound O （√τL） for the well-known Nesterov-Todd search direction and O （τL） for the xs and sx search directions.
基金Sponsored by the National Natural Science Foundation of China(Grant No.11461021)the Natural Science Basic Research Plan in Shaanxi Province of China(Grant No.2017JM1014)Scientific Research Project of Hezhou University(Grant Nos.2014YBZK06 and 2016HZXYSX03)
文摘This paper proposes a new full Nesterov-Todd(NT) step infeasible interior-point algorithm for semidefinite programming. Our algorithm uses a specific kernel function, which is adopted by Liu and Sun, to deduce the feasibility step. By using the step, it is remarkable that in each iteration of the algorithm it needs only one full-NT step, and can obtain an iterate approximate to the central path. Moreover, it is proved that the iterative bound corresponds with the known optimal one for semidefinite optimization problems.
基金supported by the National Natural Science Foundation of China (Grant No.10771133)the Shanghai Leading Academic Discipline Project (Grant No.S30101)the Research Foundation for the Doctoral Program of Higher Education (Grant No.200802800010)
文摘The choice of self-concordant functions is the key to efficient algorithms for linear and quadratic convex optimizations, which provide a method with polynomial-time iterations to solve linear and quadratic convex optimization problems. The parameters of a self-concordant barrier function can be used to compute the complexity bound of the proposed algorithm. In this paper, it is proved that the finite barrier function is a local self-concordant barrier function. By deriving the local values of parameters of this barrier function, the desired complexity bound of an interior-point algorithm based on this local self-concordant function for linear optimization problem is obtained. The bound matches the best known bound for small-update methods.
文摘In this paper, we propose a new infeasible interior-point algorithm with full NesterovTodd (NT) steps for semidefinite programming (SDP). The main iteration consists of a feasibility step and several centrality steps. We used a specific kernel function to induce the feasibility step. The analysis is more simplified. The iteration bound coincides with the currently best known bound for infeasible interior-point methods.
基金supported by the National Natural Science Foundation of China (Grant No.10771133)the Shanghai Pujiang Program (Grant No.06PJ14039)
文摘A polynomial interior-point algorithm is presented for monotone linear complementarity problem （MLCP） based on a class of kernel functions with the general barrier term, which are called general kernel functions. Under the mild conditions for the barrier term, the complexity bound of algorithm in terms of such kernel function and its derivatives is obtained. The approach is actually an extension of the existing work which only used the specific kernel functions for the MLCP.
