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1.
Wen-yuSun Jin-yunYuan Ya-xiangYuan 《计算数学(英文版)》2003,21(3):295-304
In this paper we present a trust region method of conic model for linearly constrained optimization problems.We discuss trust region approaches with conic model subproblems.Some equivalent variation properties and optimality conditions are given.A trust region algorithm based on conic model is constructed.Global convergence of the method is established. 相似文献
2.
《高等学校计算数学学报》2016,(2)
The trust region(TR) method for optimization is a class of effective methods.The conic model can be regarded as a generalized quadratic model and it possesses the good convergence properties of the quadratic model near the minimizer.The Barzilai and Borwein(BB) gradient method is also an effective method,it can be used for solving large scale optimization problems to avoid the expensive computation and storage of matrices.In addition,the BB stepsize is easy to determine without large computational efforts.In this paper,based on the conic trust region framework,we employ the generalized BB stepsize,and propose a new nonmonotone adaptive trust region method based on simple conic model for large scale unconstrained optimization.Unlike traditional conic model,the Hessian approximation is an scalar matrix based on the generalized BB stepsize,which resulting a simple conic model.By adding the nonmonotone technique and adaptive technique to the simple conic model,the new method needs less storage location and converges faster.The global convergence of the algorithm is established under certain conditions.Numerical results indicate that the new method is effective and attractive for large scale unconstrained optimization problems. 相似文献
3.
In this paper, we propose and analyze a new conic trust-region algorithm for solving the unconstrained optimization problems. A new strategy is proposed to construct the conic model and the relevant conic trust-region subproblems are solved by an approximate solution method. This approximate solution method is not only easy to implement but also preserves the strong convergence properties of the exact solution methods. Under reasonable conditions, the locally linear and superlinear convergence of the proposed algorithm is established. The numerical experiments show that this algorithm is both feasible and efficient. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
4.
解线性约束优化问题的新锥模型信赖域法 总被引:1,自引:0,他引:1
本文提出了一个解线性等式约束优化问题的新锥模型信赖域方法.论文采用零空间技术消除了新锥模型子问题中的线性等式约束,用折线法求解转换后的子问题,并给出了解线性等式约束优化问题的信赖域方法.论文提出并证明了该方法的全局收敛性,并给出了该方法解线性等式约束优化问题的数值实验.理论和数值实验结果表明新锥模型信赖域方法是有效的,这给出了用新锥模型进一步研究非线性优化的基础. 相似文献
5.
Lijuan ZHAO Wenyu SUN Raimundo J. B. de SAMPAIO 《Frontiers of Mathematics in China》2014,9(5):1211-1238
We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems. 相似文献
6.
On implementing a primal-dual interior-point method for conic quadratic optimization 总被引:8,自引:0,他引:8
Based on the work of the Nesterov and Todd on self-scaled cones an implementation of a primal-dual interior-point method
for solving large-scale sparse conic quadratic optimization problems is presented. The main features of the implementation
are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type
predictor-corrector extension and sparse linear algebra to improve the computational efficiency. Finally, the implementation
exploits fixed variables which naturally occurs in many conic quadratic optimization problems. This is a novel feature for
our implementation. Computational results are also presented to document that the implementation can solve very large problems
robustly and efficiently.
Received: November 18, 2000 / Accepted: January 18, 2001 Published online: September 27, 2002
Key Words. conic optimization – interior-point methods – large-scale implementation 相似文献
7.
A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical
learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by
second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral
conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that
solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the
branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order
conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are
nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear
cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming.
We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance
capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that
conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer
programs and, hence, improving their solvability.
This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation. 相似文献
8.
Zhaocheng Cui Boying WuShaojian Qu 《Journal of Computational and Applied Mathematics》2011,235(8):2432-2441
In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems. 相似文献
9.
We propose a way to reformulate a conic system of constraints as an optimization problem. When an appropriate interior-point
method (ipm) is applied to the reformulation, the ipm iterates yield backward-approximate solutions, that is, solutions for
nearby conic systems. In addition, once the number of ipm iterations passes a certain threshold, the ipm iterates yield forward-approximate
solutions, that is, points close to an exact solution of the original conic system. The threshold is proportional to the reciprocal
of distance to ill-posedness of the original conic system.?The condition numbers of the linear equations encountered when
applying an ipm influence the computational cost at each iteration. We show that for the reformulation, the condition numbers
of the linear equations are uniformly bounded both when computing reasonably-accurate backward-approximate solutions to arbitrary
conic systems and when computing forward-approximate solutions to well-conditioned conic systems.
Received: July 11, 1997 / Accepted: August 18, 1999?Published online March 15, 2000 相似文献
10.
In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body. 相似文献
11.
Wang Chengjing 《高校应用数学学报(英文版)》2006,21(3):263-275
Trust region methods are powerful and effective optimization methods.The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods.The advantages of the above two methods can be combined to form a more powerful method for constrained optimization.The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound.At the same time,the new algorithm still possesses robust global properties.The global convergence of the new algorithm under standard conditions is established. 相似文献
12.
王承竞 《高校应用数学学报(英文版)》2006,21(3)
Trust region methods are powerful and effective optimization methods. The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods. The advantages of the above two methods can be combined to form a more powerful method for constrained optimization. The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound. At the same time, the new algorithm still possesses robust global properties. The global convergence of the new algorithm under standard conditions is established. 相似文献
13.
Gang Xu 《Journal of Differential Equations》2008,245(11):3389-3432
In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large. 相似文献
14.
In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge
of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding
an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset
of the conic. 相似文献
15.
16.
The paper considers solving of linear programming problems with p-order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints. The proposed approach is based on construction of polyhedral approximations for p-order cones, and then invoking a Benders decomposition scheme that allows for efficient solving of the approximating problems. The conducted case study of portfolio optimization with p-order conic constraints demonstrates that the developed computational techniques compare favorably against a number of benchmark methods, including second-order conic programming methods. 相似文献
17.
Hong-Quan Li 《Comptes Rendus Mathematique》2003,337(4):283-286
In this Note, we study initially the heat kernel, pt, on conic manifolds of dimension 2. Then, we improve the upper bound of pt obtained in Li (Bull. Sci. Math. 124 (2000) 365–384) on conic manifolds of dimension ?3. Finally, we study the Hölder continuity of the heat semigroup on conic manifolds. Some new phenomenons are found on conic manifolds, in particular, on conic manifolds of dimension 2. To cite this article: H.-Q. Li, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
18.
Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dual, the cone of copositive matrices. As a result of this reformulation approach, novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools, and the extensively studied cones of completely positive and of copositive matrices. In particular, this approach has been applied to solve key combinatorial optimization problems. Along this line of research, we consider polynomial optimization problems that are not necessarily quadratic. For this purpose, we use a natural extension of the cone of completely positive matrices; namely, the cone of completely positive tensors. We provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors. As a consequence of this characterization, it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems. Also, we show that the conditions underlying the characterization are conceptually the same, regardless of the degree of the polynomials defining the problem. To illustrate our results, we discuss in further detail special and relevant instances of polynomial optimization problems. 相似文献
19.
In this paper, we establish the global existence and stability of a steady symmetric shock wave for the constant supersonic
flow past an infinitely long and large curved conic body. The flow is assumed to be polytropic, isentropic and described by
a steady potential equation. Through looking for the suitable “dissipative” boundary conditions on the shock and the conic
surface together with the special form of shock equation, we show that the conic shock attached at the vertex of the cone
exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large. 相似文献
20.
Qinghong Zhang 《Optimization Letters》2010,4(3):439-447
In this paper, we present simple proofs for the main results that appear in Nunez (Math Program 91:375–390, 2002) using a
lemma in Freund and Vera (Math Program 86:225–260, 1999) for conic linear programming. Connections between interiors and boundaries
of feasible and infeasible data instances and weak and strong feasibilities of a conic linear programming primal-dual pair
are made. 相似文献