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1.
In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems with box as well as bivalent constraints. Our necessary optimality conditions can be numerically checked by solving semi-definite programming problems. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. We construct underestimators using the sum of squares convex (SOS-convex) polynomials of real algebraic geometry. An important feature of SOS-convexity that is generally not shared by the standard convexity is that whether a polynomial is SOS-convex or not can be checked by solving a semidefinite programming problem. We illustrate the versatility of our optimality conditions by simple numerical examples. 相似文献
2.
In this paper, we present Lagrange multiplier necessary conditions for global optimality that apply to non-convex optimization
problems beyond quadratic optimization problems subject to a single quadratic constraint. In particular, we show that our
optimality conditions apply to problems where the objective function is the difference of quadratic and convex functions over
a quadratic constraint, and to certain class of fractional programming problems. Our necessary conditions become necessary
and sufficient conditions for global optimality for quadratic minimization subject to quadratic constraint. As an application,
we also obtain global optimality conditions for a class of trust-region problems. Our approach makes use of outer-estimators,
and the powerful S-lemma which has played key role in control theory and semidefinite optimization. We discuss numerical examples
to illustrate the significance of our optimality conditions.
The authors are grateful to the referees for their useful comments which have contributed to the final preparation of the
paper. 相似文献
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Guoyin Li 《Journal of Optimization Theory and Applications》2012,152(3):710-726
In this paper, we establish global optimality conditions for quadratic optimization problems with quadratic equality and bivalent
constraints. We first present a necessary and sufficient condition for a global minimizer of quadratic optimization problems
with quadratic equality and bivalent constraints. Then we examine situations where this optimality condition is equivalent
to checking the positive semidefiniteness of a related matrix, and so, can be verified in polynomial time by using elementary
eigenvalues decomposition techniques. As a consequence, we also present simple sufficient global optimality conditions, which
can be verified by solving a linear matrix inequality problem, extending several known sufficient optimality conditions in
the existing literature. 相似文献
6.
We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish
tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints,
such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality
conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract
convexity. 相似文献
7.
In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable. 相似文献
8.
Z. Y. Wu J. Quan G. Q. Li J. Tian 《Journal of Optimization Theory and Applications》2012,153(2):408-435
Multivariate cubic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot
of practical applications in real world. In this paper, some necessary local optimality conditions and some necessary global
optimality conditions for cubic polynomial optimization problems with mixed variables are established. Then some local optimization
methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization
methods for cubic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local
optimality conditions and necessary global optimality conditions. A global optimization method is proposed for cubic polynomial
optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical
examples are also given to illustrate that these approaches are very efficient. 相似文献
9.
In this paper, we first establish some sufficient and some necessary global optimality conditions for quadratic integer programming
problems. Then we present a new local optimization method for quadratic integer programming problems according to its necessary
global optimality conditions. A new global optimization method is proposed by combining its sufficient global optimality conditions,
local optimization method and an auxiliary function. The numerical examples are also presented to show that the proposed optimization
methods for quadratic integer programming problems are very efficient and stable. 相似文献
10.
Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions 总被引:3,自引:0,他引:3
In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange
multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization
problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange
multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares
with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that
our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic
optimization problems.
The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South
Wales, Sydney, Australia. 相似文献
11.
In this paper, under the existence of a certificate of nonnegativity of the objective function over the given constraint set, we present saddle-point global optimality conditions and a generalized Lagrangian duality theorem for (not necessarily convex) polynomial optimization problems, where the Lagrange multipliers are polynomials. We show that the nonnegativity certificate together with the archimedean condition guarantees that the values of the Lasserre hierarchy of semidefinite programming (SDP) relaxations of the primal polynomial problem converge asymptotically to the common primal–dual value. We then show that the known regularity conditions that guarantee finite convergence of the Lasserre hierarchy also ensure that the nonnegativity certificate holds and the values of the SDP relaxations converge finitely to the common primal–dual value. Finally, we provide classes of nonconvex polynomial optimization problems for which the Slater condition guarantees the required nonnegativity certificate and the common primal–dual value with constant multipliers and the dual problems can be reformulated as semidefinite programs. These classes include some separable polynomial programs and quadratic optimization problems with quadratic constraints that admit certain hidden convexity. We also give several numerical examples that illustrate our results. 相似文献
12.
Zhiyou Wu Yongjian Yang Fusheng Bai Jing Tian 《Applied mathematics and computation》2012,218(11):6214-6231
In this paper some global optimality conditions for general quadratic {0, 1} programming problems with linear equality constraints are discussed and then some global optimality conditions for quadratic assignment problems (QAP) are presented. A local optimization method for (QAP) is derived according to the necessary global optimality conditions. A global optimization method for (QAP) is presented by combining the sufficient global optimality conditions, the local optimization method and some auxiliary functions. Some numerical examples are given to illustrate the efficiency of the given optimization methods. 相似文献
13.
In the present work, we intend to derive conditions characterizing globally optimal solutions of quadratic 0-1 programming
problems. By specializing the problem of maximizing a convex quadratic function under linear constraints, we find explicit
global optimality conditions for quadratic 0-1 programming problems, including necessary and sufficient conditions and some
necessary conditions. We also present some global optimality conditions for the problem of minimization of half-products. 相似文献
14.
In this paper, we develop the sufficient conditions for the existence of local and global saddle points of two classes of
augmented Lagrangian functions for nonconvex optimization problem with both equality and inequality constraints, which improve
the corresponding results in available papers. The main feature of our sufficient condition for the existence of global saddle
points is that we do not need the uniqueness of the optimal solution. Furthermore, we show that the existence of global saddle
points is a necessary and sufficient condition for the exact penalty representation in the framework of augmented Lagrangians.
Based on these, we convert a class of generalized semi-infinite programming problems into standard semi-infinite programming
problems via augmented Lagrangians. Some new first-order optimality conditions are also discussed.
This research was supported by the National Natural Science Foundation of P.R. China (Grant No. 10571106 and No. 10701047). 相似文献
15.
Patrick Mehlitz 《Optimization》2017,66(10):1533-1562
We consider a bilevel programming problem in Banach spaces whose lower level solution is unique for any choice of the upper level variable. A condition is presented which ensures that the lower level solution mapping is directionally differentiable, and a formula is constructed which can be used to compute this directional derivative. Afterwards, we apply these results in order to obtain first-order necessary optimality conditions for the bilevel programming problem. It is shown that these optimality conditions imply that a certain mathematical program with complementarity constraints in Banach spaces has the optimal solution zero. We state the weak and strong stationarity conditions of this problem as well as corresponding constraint qualifications in order to derive applicable necessary optimality conditions for the original bilevel programming problem. Finally, we use the theory to state new necessary optimality conditions for certain classes of semidefinite bilevel programming problems and present an example in terms of bilevel optimal control. 相似文献
16.
In this paper, we establish global necessary and sufficient optimality conditions for D.C. vector optimization problems under
reverse convex constraints. An application to vector fractional mathematical programming is also given.
Mathematics Subject Classifications (1991). Primary 90C29, Secondary 49K30. 相似文献
17.
Patrick Mehlitz 《Optimization》2016,65(6):1203-1227
This article is dedicated to the study of bilevel optimal control problems equipped with a fully convex lower level of special structure. In order to construct necessary optimality conditions, we consider a general bilevel programming problem in Banach spaces possessing operator constraints, which is a generalization of the original bilevel optimal control problem. We derive necessary optimality conditions for the latter problem using the lower level optimal value function, ideas from DC-programming and partial penalization. Afterwards, we apply our results to the original optimal control problem to obtain necessary optimality conditions of Pontryagin-type. Along the way, we derive a handy formula, which might be used to compute the subdifferential of the optimal value function which corresponds to the lower level parametric optimal control problem. 相似文献
18.
This paper studies the global optimization of polynomial programming problems using Reformulation-Linearization Technique
(RLT)-based linear programming (LP) relaxations. We introduce a new class of bound-grid-factor constraints that can be judiciously used to augment the basic RLT relaxations in order to improve the quality of lower bounds
and enhance the performance of global branch-and-bound algorithms. Certain theoretical properties are established that shed
light on the effect of these valid inequalities in driving the discrepancies between RLT variables and their associated nonlinear
products to zero. To preserve computational expediency while promoting efficiency, we propose certain concurrent and sequential
cut generation routines and various grid-factor selection rules. The results indicate a significant tightening of lower bounds,
which yields an overall reduction in computational effort for solving a test-bed of polynomial programming problems to global
optimality in comparison with the basic RLT procedure as well as the commercial software BARON. 相似文献
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The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time. 相似文献