A second derivative SQP method: theoretical issues

Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established

A second derivative SQP method: local convergence

In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ℓ1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud \ud Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud \ud Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set

A second-derivative trust-region SQP method with a "trust-region-free" predictor step

In (NAR 08/18 and 08/21, Oxford University Computing Laboratory, 2008) we introduced a second-derivative SQP method (S2QP) for solving nonlinear nonconvex optimization problems. We proved that the method is globally convergent and locally superlinearly convergent under standard assumptions. A critical component of the algorithm is the so-called predictor step, which is computed from a strictly convex quadratic program with a trust-region constraint. This step is essential for proving global convergence, but its propensity to identify the optimal active set is Paramount for recovering fast local convergence. Thus the global and local efficiency of the method is intimately coupled with the quality of the predictor step.\ud \ud In this paper we study the effects of removing the trust-region constraint from the computation of the predictor step; this is reasonable since the resulting problem is still strictly convex and thus well-defined. Although this is an interesting theoretical question, our motivation is based on practicality. Our preliminary numerical experience with S2QP indicates that the trust-region constraint occasionally degrades the quality of the predictor step and diminishes its ability to correctly identify the optimal active set. Moreover, removal of the trust-region constraint allows for re-use of the predictor step over a sequence of failed iterations thus reducing computation. We show that the modified algorithm remains globally convergent and preserves local superlinear convergence provided a nonmonotone strategy is incorporated

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems

Sensitive detection of methane at 3.3 μm using an integrating sphere and interband cascade laser

Detection of methane at 3.3μm using a DFB Interband Cascade Laser and gold coated integrating sphere is performed. A 10cm diameter sphere with effective path length of 54.5cm was adapted for use as a gas cell. A comparison between this system and one using a 25cm path length single-pass gas cell is made using direct TDLS and methane concentrations between 0 and 1000 ppm. Initial investigations suggest a limit of detection of 1.0ppm for the integrating sphere and 2.2ppm for the single pass gas cell. The system has potential applications in challenging or industrial environments subject to high levels of vibration

Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved

${\cal N} = 8$ superconformal field theories, such as the ABJM theory at Chern-Simons level $k=1$ or $2$, contain 35 scalar operators ${\cal O}_{IJ}$ with $\Delta=1$ in the ${\bf 35}_v$ representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS$_4$ gravity dual, obtained from gauged ${\cal N}=8$ supergravity, has no cubic $A^3$ couplings in its Lagrangian, where $A^{IJ}$ is the bulk dual of ${\cal O}_{IJ}$. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS$_4$ boundary requires the introduction of a finite $A^3$ counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the $\Delta=1$ operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.Comment: 67 page

Integrating Color, Texture, and Geometry for Image Retrieval

This paper examines the problem of image retrieval from large, heterogeneous image databases. We present a technique that fulfills several needs identified by surveying recent research in the field. This technique fairly integrates a diverse and expandable set of image properties (for example, color, texture, and location) in a retrieval framework, and allows end-users substantial control over their use. We propose a novel set of evaluation methods in addition to applying established tests for image retrieval; our technique proves competitive with state-of-the-art methods in these tests and does better on certain tasks. Furthermore, it improves on many standard image retrieval algorithms by supporting queries based on subsections of images. For certain queries this capability significantly increases the relevance of the images retrieved, and further expands the user’s control over the retrieval process

Synchronous gastric and duodenal metastases from head and neck squamous cell carcinoma: a unique presentation of upper gastrointestinal bleeding.

Metastatic disease to the stomach or duodenum is an infrequent diagnosis, and head and neck squamous cell carcinoma (HNSCC) is one of the least common primary malignancies that lead to gastric or duodenal metastases. We report the case of a 65-year-old man with human immunodeficiency virus infection and previously diagnosed HNSCC who presented with melena. The patient had a percutaneous endoscopic gastrostomy tube placed 3 months prior to his presentation. Laboratory testing was significant for normocytic anemia and a digital rectal examination was positive for melena. Esophagogastroduodenoscopy revealed numerous cratered nodules with contact bleeding in the stomach as well as the duodenum that appeared malignant. Biopsies of the gastric and duodenal nodules were positive for p40 and CK 5/6, consistent with metastatic squamous cell carcinoma