Degeneracy in NLP and the development of results motivated by its presence

We study notions of nondegeneracy and several levels of increasing degeneracy from the perspective of the local behavior of a local solution of a nonlinear program when problem parameters are slightly perturbed. Ideal nondegeneracy at a local minimizer is taken to mean satisfaction of second order sufficient conditions, linear independence and strict complimentary slackness. Following a brief exploration of the relationship of these conditions with the classical definition of nondegeneracy in linear programming, we recall a number of optimality and regularity conditions used to attempt to resolve degeneracy and survey results of Fiacco, McCormick, Robinson, Kojima, Gauvin and Janin, Shapiro, Kyparisis and Liu. This overview may be viewed as a structured survey of sensitivity and stability results: the focus is on progressive levels of degeneracy. We note connections of nondegeneracy with the convergence of algorithms and observe the striking parallel between the effects of nondegeneracy and degeneracy on optimality conditions, stability analysis and algorithmic convergence behavior. Although our orientation here is primarily interpretive and noncritical, we conclude that more effort is needed to unify optimality, stability and convergence theory and more results are needed in all three areas for radically degenerate problems.Research supported by National Science Foundation Grant ECS 90-00560 and Grant N00014-89-J-1537 Office of Naval Research     

Sensitivity and stability analysis for nonlinear programming

We give a brief overview of important results in several areas of sensitivity and stability analysis for nonlinear programming, focusing initially on qualitative characterizations (e.g., continuity, differentiability and convexity) of the optimal value function. Subsequent results concern quantitative measures, in particular optimal value and solution point parameter derivative calculations, algorithmic approximations, and bounds. Our treatment is far from exhaustive and concentrates on results that hold for smooth well-structured problems.Research supported by National Science Foundation Grant ECS-86-19859 and Grant N00014-89-J-1537 Office of Naval Research.     

Perturbed variations of penalty function methods

Penalty function techniques are well known perturbation methods for solving mathematical programming problems. We define new classes of penalty functions by introducing simple perturbations of classical penalty functions or, equivalently, perturbations of the given problem. Motivation is a recently developed method called Projective SUMT, proposed by McCormick, based on solving the differential equation associated with a barrier function minimizing trajectory. We show that this trajectory-following algorithm is a simple variation of classical SUMT (Sequential Unconstrained Minimization Technique). This leads to numerous additional interpretations, simplified convergence results, duality relationships and extensions. Like SUMT, Projective SUMT is closely related to the approach of Karmarkar.Research supported by Grant ECS-86195859 and NSF N00014-85-K-0052, Office of Naval Research.     

Polysialylated-neural cell adhesion molecule (PSA-NCAM) in the human trigeminal ganglion and brainstem at prenatal and adult ages

Background  

The polysialylated neuronal cell adhesion molecule (PSA-NCAM) is considered a marker of developing and migrating neurons and of synaptogenesis in the immature vertebrate nervous system. However, it persists in the mature normal brain in some regions which retain a capability for morphofunctional reorganization throughout life. With the aim of providing information relevant to the potential for dynamic changes of specific neuronal populations in man, this study analyses the immunohistochemical occurrence of PSA-NCAM in the human trigeminal ganglion (TG) and brainstem neuronal populations at prenatal and adult age.     

Suggested research topics in sensitivity and stability analysis for semi-infinite programming problems

We suggest several important research topics for semi-infinite programs whose problem functions and index sets contain parameters that are subject to perturbation. These include optimal value and optimal solution sensitivity and stability properties and penalty function approximation techniques. The approaches proposed are a natural carryover from parametric nonlinear programming, with emphasis on practical applicability and computability.Research supported by National Science Foundation Grant SES 8722504 and Grant ECS-86-19859 and Grant N00014-89-J-1537, Office of Naval Research.     

Generalized convexity and concavity of the optimal value function in nonlinear programming

In this paper we consider generalized convexity and concavity properties of the optimal value functionf ^* for the general parametric optimization problemP(_ε) of the form min _x f(x, ε) s.t.x∈R(ε). Many results on convexity and concavity characterizations off ^* were presented by the authors in a previous paper. Such properties off ^* and the solution set mapS ^* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. We give sufficient conditions for several types of generalized convexity and concavity off ^*, in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point-to-set mapR. Specializations of these results to the parametric inequality-equality constrained nonlinear programming problem are provided. Research supported by Grant ECS-8619859, National Science Foundation and Contract N00014-86-K-0052, Office of Naval Research.     

Microwave investigation of sulfuric acid monohydrate

The complex H2SO4-H2O has been observed by rotational spectroscopy in a supersonic jet. A-type spectra for 18 isotopic forms have been analyzed, and the vibrationally averaged structure of the system has been determined. The complex forms a distorted, six-membered ring with the water unit acting as both a hydrogen bond donor and a hydrogen bond acceptor toward the sulfuric acid. One of the H2SO4 protons forms a short, direct hydrogen bond to the water oxygen, with an H...O distance of 1.645(5) A and an O-H...O angle of 165.2(4) degrees. Additionally, the orientation of the water suggests a weaker, secondary hydrogen bond between one of the H2O hydrogens and a nearby S=O oxygen on the sulfuric acid, with an O...H distance of 2.05(1) A and an O-H...O angle of 130.3(5) degrees. The experimentally determined structure is in excellent agreement with previously published DFT studies. Experiments with HOD in the jet reveal the formation of only isotopomers involving deuterium in the secondary hydrogen bond, providing direct experimental evidence for the secondary H...O interaction. Extensive isotopic substitution has also permitted a re-determination of the structure of the H2SO4 unit within the complex. The hydrogen-bonding OH bond of the sulfuric acid elongates by 0.07(2) A relative to that in free H2SO4, and the S=O bond involved in the secondary interaction stretches by 0.04(1) A. These changes reflect substantial distortion of the H2SO4 moiety in response to only a single water molecule, and prior to the proton transfer event. Spectral data indicate that the complex undergoes at least one, and probably more than one type of internal motion. Although the sulfuric acid in this work was produced from direct reaction of SO3 and water in the jet, experiments with H2(18)O indicate that about 2-3% of the acid is formed via processes not normally associated with the gas-phase hydration of SO3.     

Structure, bonding, and dipole moment of (CH3)3N-SO3. A microwave study

Six isotopic derivatives of the complex (CH3)3N-SO3 have been studied in the gas phase by microwave spectroscopy. The N-S bond length is 1.912(20) A, and the NSO angle is 100.1(2) degrees. The dipole moment, determined from Stark effect measurements, is 7.1110(69) D, representing an enhancement of 6.5 D over the sum of the dipole moments of the free monomers. Analysis of the 14N nuclear hyperfine structure indicates that about 0.6 e is transferred from the nitrogen to the SO3 upon formation of the complex. Comparison between the gas-phase structure and that previously determined for the adduct in the solid state reveals small but significant differences, indicating that the formation of the dative bond is slightly less advanced in the gas. Gas-phase and solid-state structural data are compared for several related amine-SO3 systems.     

Convergence properties of local solutions of sequences of mathematical programming problems in general spaces

This paper gives several sets of sufficient conditions that alocal solutionx ^k exists of the problem \(\min _{R^k } f^k (x)\) ,k=1, 2,..., such that {x ^k } has cluster points that arelocal solutions of a problem of the form min _R f(x). The results are based on a well-known concept of topological, orpoint-wise convergence of the sets {R ^k } toR. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis and provide additional characterizations of optimality for large classes of nonlinear programming problems.     

Screening and confirmation analysis of stimulants, narcotics and beta-adrenergic agents in human urine by hydrophilic interaction liquid chromatography coupled to mass spectrometry

The chromatographic behaviour of 44 polar compounds (23 beta-adrenergic agents, 11 stimulants, 4 narcotics and 6 phenolalkylamines) included in the list of prohibited substances and methods of the World Anti-Doping Agency, has been investigated under hydrophilic interaction liquid chromatography conditions by application of different mobile phase compositions (percentage of the organic solvent, type and amount of mobile phase additive and ionic strength) and column temperatures. Detection of analytes was performed by a triple quadrupole mass spectrometer in positive ionization mode and selected reaction monitoring acquisition mode after liquid/liquid extraction. Data collected using as stationary phase type-B silica materials from different producers, showed that the best chromatographic conditions in terms of peak shape, selectivity and chromatographic retention were obtained using an initial percentage of acetonitrile of 90%, a column temperature of 35 °C, a mobile phase pH of 4.5 and ammonium acetate (5 mM) and acetic acid (0.1%) as mobile phase additives. The selected chromatographic conditions were used to develop screening and confirmation analytical procedures to detect polar compounds in human urine for antidoping purpose. The developed methods were validated in terms of specificity, matrix effect, linearity, precision, accuracy, sensitivity, robustness and repeatability of retention times and relative ion abundances. Such methods offer attractive alternatives and considerable advantages over traditional approaches especially for the analysis of the phenolalkylamines.