A Newton-bracketing method for a simple conic optimization problem

For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs), we propose a Newton-bracketing method to improve the performance of the bisection-projection method implemented in BBCPOP [to appear in ACM Tran. Softw., 2019]. The relaxation problem is converted into the problem of finding the largest zero $y^*$ of a continuously differentiable (except at $y^*$) convex function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that $g(y) = 0$ if $y \leq y^*$ and $g(y) > 0$ otherwise. In theory, the method generates lower and upper bounds of $y^*$ both converging to $y^*$. Their convergence is quadratic if the right derivative of $g$ at $y^*$ is positive. Accurate computation of $g'(y)$ is necessary for the robustness of the method, but it is difficult to achieve in practice. As an alternative, we present a secant-bracketing method. We demonstrate that the method improves the quality of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large scale QAP instances from QAPLIB are reported.Comment: 19 pages, 2 figure

The mathematics of object recognition in machine and human vision

The purpose of this project was to see why projective geometry is related to the sort of sensors that machines and humans use for vision

A Cart, a box, a GPS: A Luggage cart and a clip style information device design from the view of universal design

The existing design of the airport luggage cart, which is intended to help travelers carry multiples pieces of luggage, has some issues. Also, the travelers sometimes are challenged to get information or to communicate with the workers at the airports especially when people travel outside of their mother country. These issues show that the airport needs a new luggage cart that is designed under the aspect of Universal Design. Therefore, this study proposed a new luggage cart design and the possibility of it to provide better service for all

Junctions of the mass-deformed nonlinear sigma models on SO(2N)/U(N)SO(2N)/U(N) and Sp(N)/U(N)Sp(N)/U(N)

We holomorphically embed nonlinear sigma models (NLSMs) on $SO(2N)/U(N)$ and $Sp(N)/U(N)$ in the hyper-K\"{a}hler (HK) NLSM on the cotangent bundle of the Grassmann manifold $T^\ast G_{2N,N}$, which is defined by $G_{N+M,M}=\frac{SU(N+M)}{SU(N)\times SU(M)\times U(1)}$, in the ${\mathcal{N}}=1$ superspace formalism and construct three-pronged junctions of the mass-deformed NLSMs (mNLSMs) in the moduli matrix formalism (MMF) by using a recently proposed method.Comment: 10 pages, 2 figure

Strong duality of a conic optimization problem with a single hyperplane and two cone constraints

Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view of strong duality of COPs for the known results. For our framework, we employ a COP minimizing a linear function in a vector variable $x$ subject to a single hyperplane constraint $x \in H$ and two cone constraints $x \in K_1$, $x \in K_2$. It can be identically reformulated as a simpler COP with the single hyperplane constraint $x \in H$ and the single cone constraint $x \in K_1 \cap K_2$. This simple COP and its dual as well as their duality relation can be represented geometrically, and they have no duality gap without any constraint qualification. The dual of the original target COP is equivalent to the dual of the reformulated COP if the Minkowski sum of the duals of the two cones $K_1$ and $K_2$ is closed or if the dual of the reformulated COP satisfies a certain Slater condition. Thus, these two conditions make it possible to transfer all duality results, including the existence and/or boundedness of optimal solutions, on the reformulated COP to the ones on the original target COP, and further to the ones on a standard primal-dual pair of COPs with symmetry