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

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

Bis(μ2-4,7-dimethyl-4,7-diazadecane-1,10-dithiolato)trinickel(II) bis(perchlorate)

In the title compound, [Ni3(C10H22N2S2)2](ClO4)2, the complex cation consists of a nickel(II) ion and two [Ni(C10H22N2S2)] units with an N2S2 tetra­dentate ligand, 3,3′-[1,2-ethane­diylbis(methyl­imino)]bis­(1-propane­thiol­ate). The central NiII ion is located on a crystallographic inversion centre and is bound to the four S atoms of the two [Ni(C10H22N2S2)] units to form a linear sulfur-bridged trimetallic moiety. The dihedral angle between the central NiS4 plane and the terminal NiN2S2 plane is 145.71 (5)°. In the [Ni(C10H22N2S2)] unit, the two methyl groups on the chelating N atoms are cis to each other, and the two six-membered N,S-chelate rings adopt a chair conformation. The Ni—S bond lengths and the S—Ni—S bite angles in the central NiS4 group are similar to those in the [Ni(C10H22N2S2)] unit