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1.
In this note, we consider the optimality criteria of multi-objective programming problems without constraint qualifications involving generalized convexity. Under the E-pseudoconvexity assumptions, the unified necessary and sufficient optimality conditions are established for weakly efficient and efficient solutions, respectively, in multi-objective programming problems. 相似文献
2.
In this paper, necessary optimality conditions for nonlinear programs in Banach spaces and constraint qualifications for their applicability are considered. A new optimality condition is introduced, and a constraint qualification ensuring the validity of this condition is given. When the domain space is a reflexive space, it is shown that the qualification is the weakest possible. If a certain convexity assumption is made, then this optimality condition is shown to reduce to the well-known extension of the Kuhn-Tucker conditions to Banach spaces. In this case, the constraint qualification is weaker than those previously given.This work was supported in part by the Office of Naval Research, Contract Number N00014-67-A-0321-0003 (NRO 47-095). 相似文献
3.
Given a linear transformation L:?
n
→?
n
and a matrix Q∈?
n
, where ?
n
is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?
n
+,Q) over the cone ?
n
+ of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R
n×n
, we consider the linear transformation L
A
:?
n
→?
n
defined by L
A
(X):=AX+XA
T
and show that the P- and Q-properties for L
A
are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.
Received: March 1999 / Accepted: November 1999?Published online April 20, 2000 相似文献
4.
Optimality conditions for nonconvex semidefinite programming 总被引:9,自引:0,他引:9
Anders Forsgren 《Mathematical Programming》2000,88(1):105-128
This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive
first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those
used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse.
The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with
those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained
when the constraint matrix is diagonal.
Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000 相似文献
5.
Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension
of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear
rate provided that the solutions to the primal and dual problems are nondegenerate. 相似文献
6.
We derive first- and second-order necessary optimality conditions for set-constrained optimization problems under the constraint
qualification-type conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the
so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting
constraints are given by an inclusion, with an arbitrary closed convex set on the right-hand side. Thus, for the second-order
analysis, some curvature characterizations of this set near the reference point must be taken into account.
相似文献
7.
We correct an error in the statement of Theorem 8 in [1].
Received: January 3, 2001 / Accepted: February 26, 2001?Published online May 18, 2001 相似文献
8.
黄龙光 《数学物理学报(B辑英文版)》2008,28(4):843-850
This article proposes a few tangent cones,which are relative to the constraint qualifications of optimization problems.With the upper and lower directional derivatives of an objective function,the characteristics of cones on the constraint qualifications are presented.The interrelations among the constraint qualifications,a few cones involved, and level sets of upper and lower directional derivatives are derived. 相似文献
9.
R. Henrion 《Journal of Optimization Theory and Applications》1992,72(1):187-197
The linear independence constraint qualification (LICQ) and the weaker Mangasarian-Fromovitz constraint qualification (MFCQ) are well-known concepts in nonlinear optimization. A theorem is proved suggesting that the set of feasible points for which MFCQ essentially differs from LICQ is small in a specified sense. As an auxiliary result, it is shown that, under MFCQ, the constraint set (even in semi-infinite optimization) is locally representable in epigraph form.The author wishes to thank Professor H. T. Jongen for valuable advice. 相似文献
10.
In this paper, vector equilibrium problems with constraint in Banach spaces are investigated. Kuhn–Tucker-like conditions for weakly efficient solutions are given by using the Gerstewitz’s function and nonsmooth analysis. Moreover, the sufficient conditions of weakly efficient solutions are established under the assumption of generalized invexity. As applications, necessary conditions of weakly efficient solutions for vector variational inequalities with constraint and vector optimization problems with constraint are obtained. 相似文献
11.
This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R 0-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically. 相似文献
12.
This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R0-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically. 相似文献
13.
Constraint qualifications in terms of approximate Jacobians are investigated for a nonsmooth constrained optimization problem, in which the involved functions are continuous but not necessarily locally Lipschitz. New constraint qualifications in terms of approximate Jacobians, weaker than the generalized Robinson constraint qualification (GRCQ) in Jeyakumar and Yen [V. Jeyakumar, N.D. Yen, Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization, SIAM J. Optim. 14 5 (2004) 1106-1127], are introduced and some examples are provided to show the utility of constrained qualifications introduced. Since the calmness condition is regarded as the basic condition for optimality conditions, the relationships between the constraint qualifications proposed and the calmness of solution mapping are also studied. 相似文献
14.
In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint qualifications, the closed cone constraint qualification for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint qualification for vector-valued quasiconvex programming (the VQ-BCQ). Based on previous results by Benoist et al. (Proc Am Math Soc 13:1109–1113, 2002), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554–563, 2011), and (Nonlinear Anal 74:1279–1285, 2011), we show that the VQ-CCCQ (resp. the VQ-BCQ) is the weakest constraint qualification for Lagrangian-type strong (resp. min–max) duality. As consequences of the main results, we study semi-definite quasiconvex programming problems in our scheme, and we observe the weakest constraint qualifications for Lagrangian-type strong and min–max dualities. Finally, we summarize the characterizations of the weakest constraint qualifications for convex and quasiconvex programming. 相似文献
15.
Mathematical Programming - Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear... 相似文献
16.
Numerical Algorithms - In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the... 相似文献
17.
We consider semidefinite monotone linear complementarity problems (SDLCP) in the space n of real symmetric n×n-matrices equipped with the cone n+ of all symmetric positive semidefinite matrices. One may define weighted (using any Mn++ as weight) infeasible interior point paths by replacing the standard condition XY=rI, r>0, (that defines the usual central path) by (XY+YX)/2=rM. Under some mild assumptions (the most stringent is the existence of some strictly complementary solution of (SDLCP)), these paths have a limit as r0, and they depend analytically on all path parameters (such as r and M), even at the limit point r=0.Mathematics Subject Classification (1991): 90C33, 65K05 相似文献
18.
Mathematical Programming - In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear... 相似文献
19.
In this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one. When the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity, such conditions collapse into the well known constraint qualifications. 相似文献
20.
We consider a nonsmooth multiobjective programming problem with inequality and set constraints. By using the notion of convexificator, we extend the Abadie constraint qualification, and derive the strong Kuhn-Tucker necessary optimality conditions. Some other constraint qualifications have been generalized and their interrelations are investigated. 相似文献