We investigate the focusing Ḣ-critical nonlinear Schrödinger equation (NLS) of Hartree type i∂tu + ∆u = −(| · |−3 ∗ |u|2)u with Ḣ radial data in dimension d = 5. It is proved that if the maximal life-span solution obeys supt ∥|∇| 12 u ∥∥ 2 < √ 6 3 ∥|∇| 12Q ∥∥ 2 , where Q is the positive radial solution to the elliptic equation with nonlocal operator (1.4) which corresponds to a new variational ...

We consider the problem of minimizing an indefinite quadratic objective function subject to twosided indelinite quadratic constraints. Under a suitable simultaneous diagonalization assumption {which trivially holds for trust region type problems), we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints. We then consider a special problem ...

The network localization problem with convex and non-convex distance constraints may be modeled as a nonlinear optimization problem. The existing localization techniques are mainly based on convex optimization. In those techniques, the non-convex distance constraints are either ignored or relaxed into convex constraints for using the convex optimization methods like SDP, least square approximat...

We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise-...

My thesis focuses on global optimization of nonconvex integral objective functions subject to parameter dependent ordinary differential equations. In particular, efficient, deterministic algorithms are developed for solving problems with both linear and nonlinear dynamics embedded. The techniques utilized for each problem classification are unified by an underlying composition principle transfe...

In this paper, we first characterize the convex $L$-subgroup of an $L$-ordered group by means of fourkinds of cut sets of an $L$-subset. Then we consider the homomorphic preimages and the product of convex $L$-subgroups.After that, we introduce an $L$-convex structure constructed by convex $L$-subgroups.Furthermore, the notion of the degree to which an $L$-subset of an $L$-ord...