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991.
Let be a semialgebraic set defined by multivariate polynomials g
i
(x). Assume S is convex, compact and has nonempty interior. Let , and ∂ S (resp. ∂ S
i
) be the boundary of S (resp. S
i
). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set
S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from
Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia,
1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International
Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of , does there exist an LMI representable set Ŝ in some higher dimensional space whose projection down onto equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following
are the main contributions of this paper: (i) assume g
i
(x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem
of minimizing any nonzero linear function ℓ
T
x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g
i
(x) is either sos-concave ( − ∇2
g
i
(x) = W(x)
T
W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S
i
is either sos-convex or poscurv-convex (S
i
is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g
i
(x) ≠ 0 on ∂ S
i
∩ S), then S is SDP representable. This also holds for S
i
for which ∂ S
i
∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).
相似文献
992.
In this paper, the approximation properties of q-Durrmeyer operators Dn,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process limn→∞Dn,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators Dn,q. 相似文献
993.
994.
John A.D. Appleby Ma?gorzata Guzowska Alexandra Rodkina 《Applied mathematics and computation》2010,217(2):763-774
We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability. 相似文献
995.
If the Aubry set satisfies some topological hypothesis, such as H1(M×T,A(c),R)≠0, then the α-function has a flat. In this paper, we will prove that has infinitely many -minimal homoclinic orbits when c′ is on the boundary of the maximal flat of the α-function. These homoclinic orbits are different from the usually called multi-bump solutions. 相似文献
996.
In the paper, we consider the following diffusive prey-predator model:
997.
Irena Rachnkov 《Journal of Mathematical Analysis and Applications》2006,320(2):611-618
We study singular boundary value problems with mixed boundary conditions of the form where , , f is a nonnegative function and satisfies the Carathéodory conditions on . Here, f can have a time singularity at t=0 and/or t=T and a space singularity at x=0 and/or y=0. We present conditions for the existence of solutions positive on [0,T) and having continuous first derivatives on [0,T]. 相似文献
998.
In this paper we consider the existence of positive solutions of the following boundary value problem:
999.
Yuhong Ma 《Journal of Mathematical Analysis and Applications》2006,314(1):97-108
We show that there exists at least one positive solution for Lidstone boundary value problem
1000.
Yuji Liu 《Journal of Mathematical Analysis and Applications》2006,323(1):721-740
The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations