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1.
The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems 总被引:1,自引:0,他引:1
The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=g−h (with g,h being lower semicontinuous proper convex functions on R
n
) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different
and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to
be more robust and more efficient than related standard methods, especially in the large scale setting. The computational
efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs
(when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient
to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization
techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization
problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds
new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported. 相似文献
2.
A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When
M is a P-matrix, the sequence {ϑ(x
k, λ
k)} (where {(x
k, λ
k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations
if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising.
Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171 相似文献
3.
《Optimization》2012,61(5):757-773
In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P 0 matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article. 相似文献
4.
We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is
instead of reducing to obtain the usualAD
2
A
T system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the productAD
2
A
T whenA has dense columns; and (2) it can easily be used to solve nonseparable quadratic programs since it requires only thatD be symmetric. We also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense. 相似文献
5.
This paper investigate a class of semi-supervised support vector machines ( $\text{ S }^3\mathrm{VMs}$ ) with arbitrary norm. A general framework for the $\text{ S }^3\mathrm{VMs}$ was first constructed based on a robust DC (Difference of Convex functions) program. With different DC decompositions, DC optimization formulations for the linear and nonlinear $\text{ S }^3\mathrm{VMs}$ are investigated. The resulting DC optimization algorithms (DCA) only require solving simple linear program or convex quadratic program at each iteration, and converge to a critical point after a finite number of iterations. The effectiveness of proposed algorithms are demonstrated on some UCI databases and licorice seed near-infrared spectroscopy data. Moreover, numerical results show that the proposed algorithms offer competitive performances to the existing $\text{ S }^3\mathrm{VM}$ methods. 相似文献
6.
Affine generalized Nash equilibrium problems (AGNEPs) represent a class of non-cooperative games in which players solve convex quadratic programs with a set of (linear) constraints that couple the players’ variables. The generalized Nash equilibria (GNE) associated with such games are given by solutions to a linear complementarity problem (LCP). This paper treats a large subclass of AGNEPs wherein the coupled constraints are shared by, i.e., common to, the players. Specifically, we present several avenues for computing structurally different GNE based on varying consistency requirements on the Lagrange multipliers associated with the shared constraints. Traditionally, variational equilibria (VE) have been amongst the more well-studied GNE and are characterized by a requirement that the shared constraint multipliers be identical across players. We present and analyze a modification to Lemke’s method that allows us to compute GNE that are not necessarily VE. If successful, the modified method computes a partial variational equilibrium characterized by the property that some shared constraints are imposed to have common multipliers across the players while other are not so imposed. Trajectories arising from regularizing the LCP formulations of AGNEPs are shown to converge to a particular type of GNE more general than Rosen’s normalized equilibrium that in turn includes a variational equilibrium as a special case. A third avenue for constructing alternate GNE arises from employing a novel constraint reformulation and parameterization technique. The associated parametric solution method is capable of identifying continuous manifolds of equilibria. Numerical results suggest that the modified Lemke’s method is more robust than the standard version of the method and entails only a modest increase in computational effort on the problems tested. Finally, we show that the conditions for applying the modified Lemke’s scheme are readily satisfied in a breadth of application problems drawn from communication networks, environmental pollution games, and power markets. 相似文献
7.
The concept of multitasking mathematical programs is discussed, and an application of multitasking to the multiple-cost-row linear programming problem is considered. Based on this, an algorithm for solving the Linear Complementarity Problem (LCP) in parallel is presented. A variety of computational results are presented using this multitasking approach on the CRAY X-MP/48. These results were obtained for randomly generated LCP's where thenxn dense matrixM has no special properties (hence, the problem is NP-hard). based on these results, an average time performance ofO(n
4) is observed. 相似文献
8.
Rita Meyer-Spasche 《Numerische Mathematik》1972,19(5):433-438
Summary This paper describes a method of solving the Liapounov equation (1)HM+M
*
H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M
*
A
*
=2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH
j
andD
j
respectively for whichH
j
M+M
*
H
j
=2D
j
is valid. Then one can solve the real system
to obtain the solution
of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom. 相似文献
9.
My master thesis concerns the solution linear complementarity problems (LCP). The Lemke algorithm, the most commonly used algorithm for solving a LCP until this day, was compared with the piecewise Newton method (PLN algorithm). The piecewise Newton method is an algorithm to solve a piecewise linear system on the basis of damped Newton methods. The linear complementarity problem is formulated as a piecewise linear system for the applicability of the PLN algorithm. Then, different application examples will be presented, solved with the PLN algorithm. As a result of the findings (of my master thesis) it can be assumed that – under the condition of coherent orientation – the PLN-algorithm requires fewer iterations to solve a linear complementarity problem than the Lemke algorithm. The coherent orientation for piecewise linear problems corresponds for linear complementarity problems to the P-matrix-property. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
Some new properties of the Projection DC decomposition algorithm (we call it Algorithm A) and the Proximal DC decomposition algorithm (we call it Algorithm B) Pham Dinh et al. in Optim Methods Softw, 23(4): 609–629 (2008) for solving the indefinite quadratic
programming problem under linear constraints are proved in this paper. Among other things, we show that DCA sequences generated
by Algorithm A converge to a locally unique solution if the initial points are taken from a neighborhood of it, and DCA sequences
generated by either Algorithm A or Algorithm B are all bounded if a condition guaranteeing the solution existence of the given
problem is satisfied. 相似文献
11.
We address a class of particularly hard-to-solve combinatorial optimization problems, namely that of multicommodity network optimization when the link cost functions are discontinuous step increasing. Unlike usual approaches consisting in the development of relaxations for such problems (in an equivalent form of a large scale mixed integer linear programming problem) in order to derive lower bounds, our d.c.(difference of convex functions) approach deals with the original continuous version and provides upper bounds. More precisely we approximate step increasing functions as closely as desired by differences of polyhedral convex functions and then apply DCA (difference of convex function algorithm) to the resulting approximate polyhedral d.c. programs. Preliminary computational experiments are presented on a series of test problems with structures similar to those encountered in telecommunication networks. They show that the d.c. approach and DCA provide feasible multicommodity flows x
* such that the relative differences between upper bounds (computed by DCA) and simple lower bounds r:=(f(x*)-LB)/{f(x*)} lies in the range [4.2 %, 16.5 %] with an average of 11.5 %, where f is the cost function of the problem and LB is a lower bound obtained by solving the linearized program (that is built from the original problem by replacing step increasing cost functions with simple affine minorizations). It seems that for the first time so good upper bounds have been obtained. 相似文献
12.
Keisuke Hotta Masatora Inaba Akiko Yoshise 《Computational Optimization and Applications》2000,17(2-3):183-201
We consider the standard linear complementarity problem (LCP): Find (x, y) R
2n
such that y = M
x + q, (x, y) 0 and x
i
y
i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P
0 LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in
Newton iterations where
is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods. 相似文献
13.
A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem
is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the
lowest-dimensional space
with the largest time step Δt
1; subsequent approximations are generated on a succession of higher-dimensional spaces
with small time step Δt
j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented
for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal
accuracy of the approximate solution in H
1-norm and L
2-norm are investigated, i.e., m
j∼m
j−1
3/2
, Δt
j∼Δt
j−1
3/2
, j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method.
Mathematics subject classifications (2000) 35L70, 65N30, 76D06
Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City
University of Hong Kong Research Project 7001093, NSF of China 50323001. 相似文献
14.
This paper addresses a new continuous approach based on the DC (Difference of Convex functions) programming and DC algorithms
(DCA) to Binary quadratic programs (BQP) which play a key role in combinatorial optimization. DCA is completely different
from other avalaible methods and featured by generating a convergent finite sequence of feasible binary solutions (obtained
by solving linear programs with the same constraint set) with decreasing objective values. DCA is quite simple and inexpensive
to handle large-scale problems. In particular DCA is explicit, requiring only matrix-vector products for Unconstrained Binary
quadratic programs (UBQP), and can then exploit sparsity in the large-scale setting. To check globality of solutions computed
by DCA, we introduce its combination with a customized Branch-and-Bound scheme using DC/SDP relaxation. The combined algorithm
allows checking globality of solutions computed by DCA and restarting it if necessary and consequently accelerates the B&B
approach. Numerical results on several series test problems provided in OR-Library (Beasley in J Global Optim, 8:429–433,
1996), show the robustness and efficiency of our algorithm with respect to standard methods. In particular DCA provides ϵ-optimal
solutions in almost all cases after only one restarting and the combined DCA-B&B-SDP always provides (ϵ−)optimal solutions. 相似文献
15.
In this paper, we introduce a construction method of total ordering cone on
\mathbbRn{\mathbb{R}^n} . It is shown that any total ordering cone on
\mathbbRn{\mathbb{R}^n} is isomorphic to the cone
\mathbbRnlex{\mathbb{R}^n_{lex}} . Existence of a total ordering cone that contain given cone with a compact base is shown. By using this cone, a solving
method of vector and set valued optimization problems is presented. 相似文献
16.
In this article, we introduce two new asynchronous multisplitting methods for solving the system of weakly nonlinear equations
Ax = G(x) in which A is an n × n real matrix and G(x) = (g
1(x), g
2(x), . . . , g
n
(x))
T
is a P-bounded mapping. First, by generalized accelerated overrelaxation (GAOR) technique, we introduce the asynchronous parallel
multisplitting GAOR method (including the synchronous parallel multisplitting AOR method as a special case) for solving the
system of weakly nonlinear equations. Second, asynchronous parallel multisplitting method based on symmetric successive overrelaxation
(SSOR) multisplitting is introduced, which is called asynchronous parallel multisplitting SSOR method. Then under suitable
conditions, we establish the convergence of the two introduced methods. The given results contain synchronous multisplitting
iterations as a special case. 相似文献
17.
Absolute value equation solution via concave minimization 总被引:3,自引:0,他引:3
O. L. Mangasarian 《Optimization Letters》2007,1(1):3-8
The NP-hard absolute value equation (AVE) Ax − |x| = b where and
is solved by a succession of linear programs. The linear programs arise from a reformulation of the AVE as the minimization of a piecewise-linear concave function on a polyhedral set and solving the latter by successive linearization. A simple MATLAB implementation of the successive linearization algorithm solved 100 consecutively generated 1,000-dimensional random instances of the AVE with only five violated equations out of a total of 100,000 equations. 相似文献
18.
The Gallant–Lambert–Vanstone (GLV) method is a very efficient technique for accelerating point multiplication on elliptic
curves with efficiently computable endomorphisms. Galbraith et al. (J Cryptol 24(3):446–469, 2011) showed that point multiplication exploiting the 2-dimensional GLV method on a large class of curves over
\mathbbFp2{\mathbb{F}_{p^2}} was faster than the standard method on general elliptic curves over
\mathbbFp{\mathbb{F}_{p}} , and left as an open problem to study the case of 4-dimensional GLV on special curves (e.g., j (E) = 0) over
\mathbbFp2{\mathbb{F}_{p^2}} . We study the above problem in this paper. We show how to get the 4-dimensional GLV decomposition with proper decomposed
coefficients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter. The resulting
implementation shows that the 4-dimensional GLV method on a GLS curve runs in about 0.78 the time of the 2-dimensional GLV
method on the same curve and in between 0.78 − 0.87 the time of the 2-dimensional GLV method using the standard method over
\mathbbFp{\mathbb{F}_{p}} . In particular, our implementation reduces by up to 27% the time of the previously fastest implementation of point multiplication
on x86-64 processors due to Longa and Gebotys (CHES2010). 相似文献
19.
Leonid G. Fel 《Functional Analysis and Other Mathematics》2011,3(2):179-192
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r12,r1r2+r12k,r3-r12k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r
1,r
2,r
3∈ℤ+, r
1≥2 and gcd(r
1,r
2)=gcd(r
1,r
3)=1, and calculate their universal Frobenius number Φ(r
1,r
2,r
3) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe
the reduction of the minimal generating sets of these semigroups up to {r12,r3-r12k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation. 相似文献
20.
We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time. 相似文献