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

Synthesis and alkyne-coupling chemistry of cyclomanganated 1- and 3-acetylindoles, 3-formylindole and analogues

The syntheses are reported of new cyclomanganated indole derivatives (1-acetyl-κO-indolyl-κC2)dicarbonylbis(trimethylphosphite)manganese (2), (1-methyl-3-acetyl-κO-indolyl-κC2)tetracarbonylmanganese (4), (3-formyl-κO-indolyl-κC2)tetracarbonylmanganese (5a) and (1-methyl-3-formyl-κO-indolyl-κC2)tetracarbonylmanganese (5b). The unusually complicated crystal structure of 5b has been determined, the first for a cyclomanganated aryl aldehyde. The preparations of a mitomycin-related pyrrolo-indole and related products by thermally promoted and oxidatively (Me3NO) initiated alkyne-coupling reactions of the previously known complex (1-acetyl-κO-indolyl-κC2)tetracarbonylmanganese (1) are reported for different alkynes and solvents. X-ray crystal structures are reported for the dimethyl acetylenedicarboxylate coupling product of 1 (dimethyl 1-methyl-l-hydroxypyrrolo[1,2a]-indole-2,3-dicarboxylate; 6a), and an unusually-cyclised triple insertion product 8 from the coupling of acetylene with 4, in which a cyclopentadiene moiety is η3-allyl-coordinated to Mn through only one double bond and an exocyclic carbon, but which rearranges on heating to an η5-cyclopentadienyl complex

Epipodial tentacle gene expression and predetermined resilience to summer mortality in the commercially important greenlip abalone, Haliotis laevigata

"Summer mortality" is a phenomenon that occurs during warm water temperature spikes that results in the mass mortality of many ecologically and economically important mollusks such as abalone. This study aimed to determine whether the baseline gene expression of abalone before a laboratory-induced summer mortality event was associated with resilience to summer mortality. Tentacle transcriptomes of 35 greenlip abalone (Haliotis laevigata) were sequenced prior to the animals being exposed to an increase in water temperature — simulating conditions which have previously resulted in summer mortality. Abalone derived from three source locations with different environmental conditions were categorized as susceptible or resistant to summer mortality depending on whether they died or survived after the water temperature was increased. We detected two genes showing significantly higher expression in resilient abalone relative to susceptible abalone prior to the laboratory-induced summer mortality event. One of these genes was annotated through the NCBI non-redundant protein database using BLASTX to an anemone (Exaiptasia pallida) Transposon Ty3-G Gag Pol polyprotein. Distinct gene expression signatures were also found between resilient and susceptible abalone depending on the population origin, which may suggest divergence in local adaptation mechanisms for resilience. Many of these genes have been suggested to be involved in antioxidant and immune-related functions. The identification of these genes and their functional roles have enhanced our understanding of processes that may contribute to summer mortality in abalone. Our study supports the hypothesis that prestress gene expression signatures are indicative of the likelihood of summer mortality

Structure of the hexameric HerA ATPase reveals a mechanism of translocation-coupled DNA-end processing in archaea.

The HerA ATPase cooperates with the NurA nuclease and the Mre11-Rad50 complex for the repair of double-strand DNA breaks in thermophilic archaea. Here we extend our structural knowledge of this minimal end-resection apparatus by presenting the first crystal structure of hexameric HerA. The full-length structure visualizes at atomic resolution the N-terminal HerA-ATP synthase domain and a conserved C-terminal extension, which acts as a physical brace between adjacent protomers. The brace also interacts in trans with nucleotide-binding residues of the neighbouring subunit. Our observations support a model in which the coaxial interaction of the HerA ring with the toroidal NurA dimer generates a continuous channel traversing the complex. HerA-driven translocation would propel the DNA towards the narrow annulus of NurA, leading to duplex melting and nucleolytic digestion. This system differs substantially from the bacterial end-resection paradigms. Our findings suggest a novel mode of DNA-end processing by this integrated archaeal helicase-nuclease machine.The SAXS data collection was supported by funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under BioStruct-X (grant agreement N°283570). Research in the N.P.R. laboratory is funded by the Medical Research Council [Career Development Award G0701443]. Research in the L.P. laboratory is funded by a Wellcome Trust Senior Fellowship Award in Basic Biomedical Sciences [grant number 08279/Z/07/Z]. Work in the L.P. and N.P.R. laboratories is also supported by an Isaac Newton Trust Research Grant, and S.M.B. is supported by a BBSRC Doctoral Training Grant.This is the author accepted manuscript. The final version is available from NPG via http://dx.doi.org/10.1038/ncomms650