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1.
For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved. 相似文献
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Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex
minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal
methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal,
and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar—Todd phenomenon
does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {||x
k
|| } and {f(x
k
) } when {x
k
} is unbounded and {f(x
k
) } is bounded for the non\-smooth minimization methods (i), (ii), and (iii).
Accepted 15 October 1996 相似文献
4.
姜爱萍 《数学物理学报(A辑)》2011,31(1):103-116
该文提出一种QP-free可行域方法用来解满足光滑不等式约束的最优化问题.此方法把QP-free方法和3-1线性互补函数相结合一个等价于原约束问题的一阶KKT条件的方程组,并在此基础上给出解这个方程组的迭代算法. 这个方法的每一步迭代都可以看作是对求KKT条件解的牛顿或拟牛顿迭代的扰动,且在该方法中每一步的迭代均具有可行性. 该方法是可实行的且具有全局性, 且不需要严格互补条件、聚点的孤立性和积极约束函数梯度的线性独立等假设. 在与文献[2]中相同的适当条件下,此方法还具有超线性收敛性. 数值检验结果表示,该文提出的QP-free可行域方法是切实有效的方法. 相似文献
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ABSTRACTRecently, a local framework of Newton-type methods for constrained systems of equations has been developed. Applied to the solution of Karush–Kuhn–Tucker (KKT) systems, the framework enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of equations. It is an open question whether a comparable result can be achieved for other (not piecewise smooth) reformulations. It will be shown that this is possible if the KKT system is reformulated by means of the Fischer–Burmeister complementarity function under conditions that allow degenerate KKT points and nonisolated Lagrange multipliers. To this end, novel constrained Levenberg–Marquardt subproblems are introduced. They allow significantly longer steps for updating the multipliers. Based on this, a convergence rate of at least 1.5 is shown. 相似文献
7.
Jane J. Ye 《Nonlinear Analysis: Theory, Methods & Applications》2012,75(3):1642-1654
The exact penalty approach aims at replacing a constrained optimization problem by an equivalent unconstrained optimization problem. Most results in the literature of exact penalization are mainly concerned with finding conditions under which a solution of the constrained optimization problem is a solution of an unconstrained penalized optimization problem, and the reverse property is rarely studied. In this paper, we study the reverse property. We give the conditions under which the original constrained (single and/or multiobjective) optimization problem and the unconstrained exact penalized problem are exactly equivalent. The main conditions to ensure the exact penalty principle for optimization problems include the global and local error bound conditions. By using variational analysis, these conditions may be characterized by using generalized differentiation. 相似文献
8.
A new method is proposed for solving box constrained global optimization problems. The basic idea of the method is described as follows: Constructing a so-called cut-peak function and a choice function for each present minimizer, the original problem of finding a global solution is converted into an auxiliary minimization problem of finding local minimizers of the choice function, whose objective function values are smaller than the previous ones. For a local minimum solution of auxiliary problems this procedure is repeated until no new minimizer with a smaller objective function value could be found for the last minimizer. Construction of auxiliary problems and choice of parameters are relatively simple, so the algorithm is relatively easy to implement, and the results of the numerical tests are satisfactory compared to other methods. 相似文献
9.
In this paper we give a variant of the Topkis—Veinott method for solving inequality constrained optimization problems. This
method uses a linearly constrained positive semidefinite quadratic problem to generate a feasible descent direction at each
iteration. Under mild assumptions, the algorithm is shown to be globally convergent in the sense that every accumulation point
of the sequence generated by the algorithm is a Fritz—John point of the problem. We introduce a Fritz—John (FJ) function,
an FJ1 strong second-order sufficiency condition (FJ1-SSOSC), and an FJ2 strong second-order sufficiency condition (FJ2-SSOSC),
and then show, without any constraint qualification (CQ), that (i) if an FJ point z satisfies the FJ1-SSOSC, then there exists a neighborhood N(z) of z such that, for any FJ point y ∈ N(z) \ {z } , f
0
(y) ≠ f
0
(z) , where f
0
is the objective function of the problem; (ii) if an FJ point z satisfies the FJ2-SSOSC, then z is a strict local minimum of the problem. The result (i) implies that the entire iteration point sequence generated by the
method converges to an FJ point. We also show that if the parameters are chosen large enough, a unit step length can be accepted
by the proposed algorithm.
Accepted 21 September 1998 相似文献
10.
Levent Tunçel 《Foundations of Computational Mathematics》2001,1(3):229-254
We generalize primal—dual interior-point methods for linear programming (LP) problems to the convex optimization problems
in conic form. Previously, the most comprehensive theory of symmetric primal—dual interior-point algorithms was given by Nesterov
and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we
allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov—Todd
algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal—dual
interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal—dual
algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication
increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better
theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric
primal—dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated
framework.
August 25, 1999. Final version received: March 7, 2001. 相似文献
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Leo Liberti 《Discrete Applied Mathematics》2009,157(6):1309-1318
This paper concerns the application of reformulation techniques in mathematical programming to a specific problem arising in quantum chemistry, namely the solution of Hartree-Fock systems of equations, which describe atomic and molecular electronic wave functions based on the minimization of a functional of the energy. Their traditional solution method does not provide a guarantee of global optimality and its output depends on a provided initial starting point. We formulate this problem as a multi-extremal nonconvex polynomial programming problem, and solve it with a spatial Branch-and-Bound algorithm for global optimization. The lower bounds at each node are provided by reformulating the problem in such a way that its convex relaxation is tight. The validity of the proposed approach was established by successfully computing the ground-state of the helium and beryllium atoms. 相似文献
15.
An application in magnetic resonance spectroscopy quantification models a signal as a linear combination of nonlinear functions. It leads to a separable nonlinear least squares fitting problem, with linear bound constraints on some variables. The variable projection (VARPRO) technique can be applied to this problem, but needs to be adapted in several respects. If only the nonlinear variables are subject to constraints, then the Levenberg–Marquardt minimization algorithm that is classically used by the VARPRO method should be replaced with a version that can incorporate those constraints. If some of the linear variables are also constrained, then they cannot be projected out via a closed-form expression as is the case for the classical VARPRO technique. We show how quadratic programming problems can be solved instead, and we provide details on efficient function and approximate Jacobian evaluations for the inequality constrained VARPRO method. 相似文献
16.
Houyuan Jiang 《Journal of Global Optimization》1996,9(2):169-181
The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition. 相似文献
17.
In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality
(SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions
of the SOCCVI problem. The first neural network uses the Fischer-Burmeister (FB) function to achieve an unconstrained minimization
which is a merit function of the Karush-Kuhn-Tucker equation. We show that the merit function is a Lyapunov function and this
neural network is asymptotically stable. The second neural network is introduced for solving a projection formulation whose
solutions coincide with the KKT triples of SOCCVI problem. Its Lyapunov stability and global convergence are proved under
some conditions. Simulations are provided to show effectiveness of the proposed neural networks. 相似文献
18.
A family of scaled conjugate gradient algorithms for large-scale unconstrained minimization is defined. The Perry, the Polak—Ribière
and the Fletcher—Reeves formulae are compared using a spectral scaling derived from Raydan's spectral gradient optimization
method. The best combination of formula, scaling and initial choice of step-length is compared against well known algorithms
using a classical set of problems. An additional comparison involving an ill-conditioned estimation problem in Optics is presented.
Accepted 22 August 2000. Online publication 26 February 2001. 相似文献
19.
In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method),
where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a
scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a
projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that
if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal
face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system
of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector
stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.}
Accepted 5 September 2000. Online publication 16 January 2001. 相似文献
20.
本文对一般非线性约束优化问题提出了一个信赖域算法,导出了等价的KKT条件.在试探步满足适当条件下,证明了算法的全局收敛性,并进行了数值试验. 相似文献