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1.
This paper introduces and analyses a new algorithm for minimizing a convex function subject to a finite number of convex inequality
constraints. It is assumed that the Lagrangian of the problem is strongly convex. The algorithm combines interior point methods
for dealing with the inequality constraints and quasi-Newton techniques for accelerating the convergence. Feasibility of the
iterates is progressively enforced thanks to shift variables and an exact penalty approach. Global and q-superlinear convergence is obtained for a fixed penalty parameter; global convergence to the analytic center of the optimal
set is ensured when the barrier parameter tends to zero, provided strict complementarity holds.
Received: December 21, 2000 / Accepted: July 13, 2001?Published online February 14, 2002 相似文献
2.
Krzysztof C. Kiwiel 《Mathematical Programming》2001,90(1):1-25
We study a general subgradient projection method for minimizing a quasiconvex objective subject to a convex set constraint
in a Hilbert space. Our setting is very general: the objective is only upper semicontinuous on its domain, which need not
be open, and various subdifferentials may be used. We extend previous results by proving convergence in objective values and
to the generalized solution set for classical stepsizes t
k
→0, ∑t
k
=∞, and weak or strong convergence of the iterates to a solution for {t
k
}∈ℓ2∖ℓ1 under mild regularity conditions. For bounded constraint sets and suitable stepsizes, the method finds ε-solutions with an
efficiency estimate of O(ε-2), thus being optimal in the sense of Nemirovskii.
Received: October 4, 1998 / Accepted: July 24, 2000?Published online January 17, 2001 相似文献
3.
Nonlinear programming without a penalty function 总被引:57,自引:0,他引:57
In this paper the solution of nonlinear programming problems by a Sequential Quadratic Programming (SQP) trust-region algorithm
is considered. The aim of the present work is to promote global convergence without the need to use a penalty function. Instead,
a new concept of a “filter” is introduced which allows a step to be accepted if it reduces either the objective function or
the constraint violation function. Numerical tests on a wide range of test problems are very encouraging and the new algorithm
compares favourably with LANCELOT and an implementation of Sl1QP.
Received: October 17, 1997 / Accepted: August 17, 2000?Published online September 3, 2001 相似文献
4.
Roman A. Polyak 《Mathematical Programming》2002,92(2):197-235
We introduce an alternative to the smoothing technique approach for constrained optimization. As it turns out for any given
smoothing function there exists a modification with particular properties. We use the modification for Nonlinear Rescaling
(NR) the constraints of a given constrained optimization problem into an equivalent set of constraints.?The constraints transformation
is scaled by a vector of positive parameters. The Lagrangian for the equivalent problems is to the correspondent Smoothing
Penalty functions as Augmented Lagrangian to the Classical Penalty function or MBFs to the Barrier Functions. Moreover the
Lagrangians for the equivalent problems combine the best properties of Quadratic and Nonquadratic Augmented Lagrangians and
at the same time are free from their main drawbacks.?Sequential unconstrained minimization of the Lagrangian for the equivalent
problem in primal space followed by both Lagrange multipliers and scaling parameters update leads to a new class of NR multipliers
methods, which are equivalent to the Interior Quadratic Prox methods for the dual problem.?We proved convergence and estimate
the rate of convergence of the NR multipliers method under very mild assumptions on the input data. We also estimate the rate
of convergence under various assumptions on the input data.?In particular, under the standard second order optimality conditions
the NR method converges with Q-linear rate without unbounded increase of the scaling parameters, which correspond to the active
constraints.?We also established global quadratic convergence of the NR methods for Linear Programming with unique dual solution.?We
provide numerical results, which strongly support the theory.
Received: September 2000 / Accepted: October 2001?Published online April 12, 2002 相似文献
5.
On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating 总被引:2,自引:0,他引:2
In previous work, the authors provided a foundation for the theory of variable metric proximal point algorithms in Hilbert
space. In that work conditions are developed for global, linear, and super–linear convergence. This paper focuses attention
on two matrix secant updating strategies for the finite dimensional case. These are the Broyden and BFGS updates. The BFGS
update is considered for application in the symmetric case, e.g., convex programming applications, while the Broyden update
can be applied to general monotone operators. Subject to the linear convergence of the iterates and a quadratic growth condition
on the inverse of the operator at the solution, super–linear convergence of the iterates is established for both updates.
These results are applied to show that the Chen–Fukushima variable metric proximal point algorithm is super–linearly convergent
when implemented with the BFGS update.
Received: September 12, 1996 / Accepted: January 7, 2000?Published online March 15, 2000 相似文献
6.
In this paper, we consider a special class of nonconvex programming problems for which the objective function and constraints
are defined in terms of general nonconvex factorable functions. We propose a branch-and-bound approach based on linear programming
relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev
interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT). A suitable partitioning process
is proposed that induces convergence to a global optimum. The algorithm has been implemented in C++ and some preliminary computational
results are reported on a set of fifteen engineering process control and design test problems from various sources in the
literature. The results indicate that the proposed procedure generates tight relaxations, even via the initial node linear
program itself. Furthermore, for nine of these fifteen problems, the application of a local search method that is initialized
at the LP relaxation solution produced the actual global optimum at the initial node of the enumeration tree. Moreover, for
two test cases, the global optimum found improves upon the solutions previously reported in the source literature.
Received: January 14, 1998 / Accepted: June 7, 1999?Published online December 15, 2000 相似文献
7.
In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP)
problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints
by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems
includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important
SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem
and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of
the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale
SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method
designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the
transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global
convergence of the algorithm is established under mild and reasonable assumptions.
Received: January 5, 2000 / Accepted: October 2001?Published online February 14, 2002 相似文献
8.
An interior Newton method for quadratic programming 总被引:2,自引:0,他引:2
We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong
convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality
conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative
interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results
of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential.
Received October 11, 1993 / Revised version received February 20, 1996
Published online July 19, 1999 相似文献
9.
Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions 总被引:5,自引:0,他引:5
In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family
of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the
polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following
algorithm of Kojima et al. and Monteiro and Adler, and the predictor-corrector algorithm of Mizuno et al., carry over to the
context of SOCP, that is they have an O( logε-1) iteration-complexity to reduce the duality gap by a factor of ε, where n is the number of second-order cones. Since the MZ-type family studied in this paper includes an analogue of the Alizadeh,
Haeberly and Overton pure Newton direction, we establish for the first time the polynomial convergence of primal-dual algorithms
for SOCP based on this search direction.
Received: June 5, 1998 / Accepted: September 8, 1999?Published online April 20, 2000 相似文献
10.
A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints 总被引:12,自引:0,他引:12
A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear
inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based
on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased.
Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The
objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling
interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity,
dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is
asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve
quadratic convergence.
Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000 相似文献
11.
Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in
which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO
as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computationally tractable robust
counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making
RO a useful tool for real-world applications. We discuss some of these applications, specifically: antenna design, truss topology
design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB
collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected
by small perturbations of the data and that the RO methodology can be successfully used to overcome this phenomenon.
Received: May 24, 2000 / Accepted: September 12, 2001?Published online February 14, 2002 相似文献
12.
Improving the robustness of descent-based methods for semismooth equations using proximal perturbations 总被引:1,自引:0,他引:1
Stephen C. Billups 《Mathematical Programming》2000,87(1):153-175
A common difficulty encountered by descent-based equation solvers is convergence to a local (but not global) minimum of an
underlying merit function. Such difficulties can be avoided by using a proximal perturbation strategy, which allows the iterates
to escape the local minimum in a controlled fashion. This paper presents the proximal perturbation strategy for a general
class of methods for solving semismooth equations and proves subsequential convergence to a solution based upon a pseudomonotonicity
assumption. Based on this approach, two sample algorithms for solving mixed complementarity problems are presented. The first
uses an extremely simple (but not very robust) basic algorithm enhanced by the proximal perturbation strategy. The second
uses a more sophisticated basic algorithm based on the Fischer-Burmeister function. Test results on the MCPLIB and GAMSLIB
complementarity problem libraries demonstrate the improvement in robustness realized by employing the proximal perturbation
strategy.
Received July 15, 1998 / Revised version received June 28, 1999?Published online November 9, 1999 相似文献
13.
We propose feasible descent methods for constrained minimization that do not make explicit use of the derivative of the objective
function. The methods iteratively sample the objective function value along a finite set of feasible search arcs and decrease
the sampling stepsize if an improved objective function value is not sampled. The search arcs are obtained by projecting search
direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone
of first-order feasible variations at the current iterate. We show that these methods have desirable convergence properties
under certain regularity assumptions on the constraints. In the case of linear constraints, the projections are redundant
and the regularity assumptions hold automatically. Numerical experience with the methods in the linearly constrained case
is reported.
Received: November 12, 1999 / Accepted: April 6, 2001?Published online October 26, 2001 相似文献
14.
A branch and cut algorithm for nonconvex quadratically constrained quadratic programming 总被引:12,自引:0,他引:12
Charles Audet Pierre Hansen Brigitte Jaumard Gilles Savard 《Mathematical Programming》2000,87(1):131-152
We present a branch and cut algorithm that yields in finite time, a globally ε-optimal solution (with respect to feasibility
and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic
terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four
classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select
the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree,
and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at
any node of the tree is flexible enough to be used at other nodes. Computational results are reported that include standard
test problems taken from the literature. Some of these problems are solved for the first time with a proof of global optimality.
Received December 19, 1997 / Revised version received July 26, 1999?Published online November 9, 1999 相似文献
15.
Song Xu 《Mathematical Programming》2000,87(3):501-517
We propose an infeasible non-interior path-following method for nonlinear complementarity problems with uniform P-functions. This method is based on the smoothing techniques introduced by Kanzow. A key to our analysis is the introduction
of a new notion of neighborhood for the central path which is suitable for infeasible non-interior path-following methods.
By restricting the iterates in the neighborhood of the central path, we provide a systematic procedure to update the smoothing
parameter and establish the global linear convergence of this method. Some preliminary computational results are reported.
Received: March 13, 1997 / Accepted: December 17, 1999?Published online February 23, 2000 相似文献
16.
A trust region method based on interior point techniques for nonlinear programming 总被引:15,自引:0,他引:15
An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential
quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the
iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but
the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is
presented.
Received: May 1996 / Accepted: August 18, 2000?Published online October 18, 2000 相似文献
17.
We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than
those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative,
and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical
regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer
than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem,
our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems
was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty
method for solving irregular problems.
Received: February 2000 / Accepted: November 2000?Published online January 17, 2001 相似文献
18.
A class of affine-scaling interior-point methods for bound constrained optimization problems is introduced which are locally
q–superlinear or q–quadratic convergent. It is assumed that the strong second order sufficient optimality conditions at the
solution are satisfied, but strict complementarity is not required. The methods are modifications of the affine-scaling interior-point
Newton methods introduced by T. F. Coleman and Y. Li (Math. Programming, 67, 189–224, 1994). There are two modifications. One is a modification of the scaling matrix, the other one is the use of a
projection of the step to maintain strict feasibility rather than a simple scaling of the step. A comprehensive local convergence
analysis is given. A simple example is presented to illustrate the pitfalls of the original approach by Coleman and Li in
the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper.
Received October 2, 1998 / Revised version received April 7, 1999?Published online July 19, 1999 相似文献
19.
Krzysztof C. Kiwiel 《Mathematical Programming》1999,85(2):241-258
k } by taking xk to be an approximate minimizer of , where is a piecewise linear model of f constructed from accumulated subgradient linearizations of f, Dh is the D-function of a generalized Bregman function h and tk>0. Convergence under implementable criteria is established by extending our recent framework of Bregman proximal minimization,
which is of independent interest, e.g., for nonquadratic multiplier methods for constrained minimization. In particular, we
provide new insights into the convergence properties of bundle methods based on h=?|·|2.
Received September 18, 1997 / Revised version received June 30, 1998
Published online November 24, 1998 相似文献
20.
Gongyun Zhao 《Mathematical Programming》2001,90(3):507-536
An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving two-stage
stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are
studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run
in polynomial-time.
Received: August 1998 / Accepted: August 2000?Published online April 12, 2001 相似文献