A Revisit to Quadratic Programming with One Inequality Quadratic Constraint via Matrix Pencil

The quadratic programming over one inequality quadratic constraint (QP1QC) is a very special case of quadratically constrained quadratic programming (QCQP) and attracted much attention since early 1990's. It is now understood that, under the primal Slater condition, (QP1QC) has a tight SDP relaxation (PSDP). The optimal solution to (QP1QC), if exists, can be obtained by a matrix rank one decomposition of the optimal matrix X? to (PSDP). In this paper, we pay a revisit to (QP1QC) by analyzing the associated matrix pencil of two symmetric real matrices A and B, the former matrix of which defines the quadratic term of the objective function whereas the latter for the constraint. We focus on the \undesired" (QP1QC) problems which are often ignored in typical literature: either there exists no Slater point, or (QP1QC) is unbounded below, or (QP1QC) is bounded below but unattainable. Our analysis is conducted with the help of the matrix pencil, not only for checking whether the undesired cases do happen, but also for an alternative way to compute the optimal solution in comparison with the usual SDP/rank-one-decomposition procedure.Comment: 22 pages, 0 figure

An SDP Approach For Solving Quadratic Fractional Programming Problems

This paper considers a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. As is well-known, the fractional objective function can be replaced by a parametric family of quadratic functions, which makes (P) highly related to, but more difficult than a single quadratic programming problem subject to a similar constraint set. The task is to find the optimal parameter $\lambda^*$ and then look for the optimal solution if $\lambda^*$ is attained. Contrasted with the classical Dinkelbach method that iterates over the parameter, we propose a suitable constraint qualification under which a new version of the S-lemma with an equality can be proved so as to compute $\lambda^*$ directly via an exact SDP relaxation. When the constraint set of (P) is degenerated to become an one-sided inequality, the same SDP approach can be applied to solve (P) {\it without any condition}. We observe that the difference between a two-sided problem and an one-sided problem lies in the fact that the S-lemma with an equality does not have a natural Slater point to hold, which makes the former essentially more difficult than the latter. This work does not, either, assume the existence of a positive-definite linear combination of the quadratic terms (also known as the dual Slater condition, or a positive-definite matrix pencil), our result thus provides a novel extension to the so-called "hard case" of the generalized trust region subproblem subject to the upper and the lower level set of a quadratic function.Comment: 26 page

On simultaneous diagonalization via congruence of real symmetric matrices

Simultaneous diagonalization via congruence (SDC) for more than two symmetric matrices has been a long standing problem. So far, the best attempt either relies on the existence of a semidefinite matrix pencil or casts on the complex field. The problem now is resolved without any assumption. We first propose necessary and sufficient conditions for SDC in case that at least one of the matrices is nonsingular. Otherwise, we show that the singular matrices can be decomposed into diagonal blocks such that the SDC of given matrices becomes equivalently the SDC of the sub-matrices. Most importantly, the sub-matrices now contain at least one nonsingular matrix. Applications to simplify some difficult optimization problems with the presence of SDC are mentioned

S-Lemma with Equality and Its Applications

Let $f(x)=x^TAx+2a^Tx+c$ and $h(x)=x^TBx+2b^Tx+d$ be two quadratic functions having symmetric matrices $A$ and $B$. The S-lemma with equality asks when the unsolvability of the system $f(x)<0, h(x)=0$ implies the existence of a real number $\mu$ such that $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$. The problem is much harder than the inequality version which asserts that, under Slater condition, $f(x)<0, h(x)\le0$ is unsolvable if and only if $f(x) + \mu h(x)\ge0, ~\forall x\in \mathbb{R}^n$ for some $\mu\ge0$. In this paper, we show that the S-lemma with equality does not hold only when the matrix $A$ has exactly one negative eigenvalue and $h(x)$ is a non-constant linear function ($B=0, b\not=0$). As an application, we can globally solve $\inf\{f(x)\vert h(x)=0\}$ as well as the two-sided generalized trust region subproblem $\inf\{f(x)\vert l\le h(x)\le u\}$ without any condition. Moreover, the convexity of the joint numerical range $\{(f(x), h_1(x),\ldots, h_p(x)):~x\in\Bbb R^n\}$ where $f$ is a (possibly non-convex) quadratic function and $h_1(x),\ldots,h_p(x)$ are affine functions can be characterized using the newly developed S-lemma with equality.Comment: 34 page

Performance estimations of first fit algorithm for online bin packing with variable bin sizes and LIB constraints

We consider the NP Hard problem of online Bin Packing while requiring that larger (or longer) items be placed below smaller (or shorter) items --- we call such a version the {LIB} version of problems. Bin sizes can be uniform or variable. We provide analytical upper bounds as well as experimental results on the asymptotic approximation ratio for the first fit algorithm

Sex Differential Genetic Effect of Chromosome 9p21 on Subclinical Atherosclerosis

BACKGROUND: Chromosome 9p21 has recently been shown to be a risk region for a broad range of vascular diseases. Since carotid intima-media thickness (IMT) and plaque are independent predictors for vascular diseases, the association between 9p21 and these two phenotypes was investigated. METHODOLOGY/PRINCIPAL FINDINGS: Carotid segment-specific IMT and plaques were obtained in 1083 stroke- and myocardial infarction-free volunteers. We tested the genotypes and haplotypes of key single nucleotide polymorphisms (SNPs) on chromosome 9p21 for the associations with carotid IMT and plaque. Multivariate permutation analyses demonstrated that carriers of the T allele of SNP rs1333040 were significantly associated with thicker common carotid artery (CCA) IMT (p=0.021) and internal carotid artery (ICA) IMT (p=0.033). The risk G allele of SNP rs2383207 was associated with ICA IMT (p=0.007). Carriers of the C allele of SNP rs1333049 were found to be significantly associated with thicker ICA IMT (p=0.010) and the greater risk for the presence of carotid plaque (OR=1.57 for heterozygous carriers; OR=1.75 for homozygous carriers). Haplotype analysis showed a global p value of 0.031 for ICA IMT and 0.115 for the presence of carotid plaque. Comparing with the other haplotypes, the risk TGC haplotype yielded an adjusted p value of 0.011 and 0.017 for thicker ICA IMT and the presence of carotid plaque respectively. Further analyzing the data separated by sex, the results were significant only in men but not in women. CONCLUSIONS: Chromosome 9p21 had a significant association with carotid atherosclerosis, especially ICA IMT. Furthermore, such genetic effect was in a gender-specific manner in the Han Chinese population