102 research outputs found
Global convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programs without constraint qualifications
In this paper, we propose a new sequential quadratic semidefinite programming
(SQSDP) method for solving nonlinear semidefinite programs (NSDPs), in which we
produce iteration points by solving a sequence of stabilized quadratic
semidefinite programming (QSDP) subproblems, which we derive from the minimax
problem associated with the NSDP. Differently from the existing SQSDP methods,
the proposed one allows us to solve those QSDP subproblems just approximately
so as to ensure global convergence. One more remarkable point of the proposed
method is that any constraint qualifications (CQs) are not required in the
global convergence analysis. Specifically, under some assumptions without CQs,
we prove the global convergence to a point satisfying any of the following: the
stationary conditions for the feasibility problem; the
approximate-Karush-Kuhn-Tucker (AKKT) conditions; the trace-AKKT conditions.
The latter two conditions are the new optimality conditions for the NSDP
presented by Andreani et al. (2018) in place of the Karush-Kuhn-Tucker
conditions. Finally, we conduct some numerical experiments to examine the
efficiency of the proposed method
Ab initio study of spin-spiral noncollinear magnetism in a free-standing Fe(110) monolayer under in-plane strain
We investigate the magnetic phase transition from collinear ferromagnetic (FM) ordering to noncollinear spin-spiral (SS) ordering in an Fe(110) monolayer under in-plane strain by performing fully unconstrained first-principles spin-density-functional calculations. The FM Fe(110) monolayer undergoes a FM-SS phase transition on the application of in-plane compression, whereas the application of tension keeps the system FM. The stability and wavelength of the excited SS state are further increased by compressive strains, especially along [ī10]. The FM-SS transition in the isotropically strained monolayer is dominated by competing exchange interactions between the ferromagnetically coupled first neighbor and the antiferromagnetically coupled second neighbor; the third neighbor also contributes to the transition under anisotropic strain. In addition, we demonstrate the stabilization mechanism of SS noncollinear magnetism from the electronic band structures: The noncollinear SS state is stabilized by a remarkable interband repulsion between the majority and minority spins, which occurs under in-plane compressio
Closing Duality Gaps of SDPs through Perturbation
Let be a primal-dual pair of SDPs with a nonzero finite
duality gap. Under such circumstances, and are weakly
feasible and if we perturb the problem data to recover strong feasibility, the
(common) optimal value function as a function of the perturbation is not
well-defined at zero (unperturbed data) since there are ``two different optimal
values'' and , where and are
the optimal values of and respectively. Thus, continuity of
is lost at zero though is continuous elsewhere. Nevertheless, we show
that a limiting version of is a well-defined monotone decreasing
continuous bijective function connecting and with
domain under the assumption that both and have
singularity degree one. The domain corresponds to directions of
perturbation defined in a certain manner. Thus, ``completely fills''
the nonzero duality gap under a mild regularity condition. Our result is tight
in that there exists an instance with singularity degree two for which
is not continuous.Comment: 26 pages. Comments welcom
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