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

An Improved Analysis of Semidefinite Approximation Bound for Nonconvex Nonhomogeneous Quadratic Optimization with Ellipsoid Constraints

We consider the problem of approximating nonconvex quadratic optimization with ellipsoid constraints (ECQP). We show some SDP-based approximation bounds for special cases of (ECQP) can be improved by trivially applying the extened Pataki's procedure. The main result of this paper is to give a new analysis on approximating (ECQP) by the SDP relaxation, which greatly improves Tseng's result [SIAM Journal Optimization, 14, 268-283, 2003]. As an application, we strictly improve the approximation ratio for the assignment-polytope constrained quadratic program.Comment: 12 page

Photoluminescence pressure coefficients of InAs/GaAs quantum dots

We have investigated the band-gap pressure coefficients of self-assembled InAs/GaAs quantum dots by calculating 17 systems with different quantum dot shape, size, and alloying profile using atomistic empirical pseudopotential method within the ``strained linear combination of bulk bands'' approach. Our results confirm the experimentally observed significant reductions of the band gap pressure coefficients from the bulk values. We show that the nonlinear pressure coefficients of the bulk InAs and GaAs are responsible for these reductions. We also find a rough universal pressure coefficient versus band gap relationship which agrees quantitatively with the experimental results. We find linear relationships between the percentage of electron wavefunction on the GaAs and the quantum dot band gaps and pressure coefficients. These linear relationships can be used to get the information of the electron wavefunctions.Comment: 8 pages, 2 tables, 4 figure

The risk factors and preventive measures for the recurrence of endometrial polyps

Endometrial polyps (EPs) are a frequently encountered gynecologic disease with abnormal uterine bleeding and infertility being the two common presenting problems, and hysteroscopic polypectomy is an effective method to remove them. The postoperative polyp recurrence might result in reappearance of abnormal uterine bleeding or infertility, whereas factors influencing the postoperative recurrence potential have limited data. Endometrial polyp recurrence remains a concern with recurrence rates of 2.5% to 43.6%. As such, it is critical to identify the risk factors and the preventive measures for endometrial recurrence, especially in reproductive-age women desiring future conception, to aid in clinical counselling and decision making. The recurrence of EPs is related to estrogen stimulation and endometrial hyperplasia. The progesterone-containing drugs are currently the most commonly used method to prevent the recurrence of EPs. In this article, authors aim to discuss the high-risk factors of EPs recurrence and the preventive measures for EPs recurrence. The preventive measures will focus on the combined oral contraceptives (COCs) and the levonorgestrel-releasing intrauterine system (LNG-IUS)