Nonsmooth Optimization and Descent Methods

Nonsmooth optimization is a field of research actively pursued at IIASA. In this paper, we show "what" it is; a thing that cannot be guessed easily from its definition by a negative statement. Also, we show "why" it exists at IIASA, by exhibiting a large field of applications ranging from the theory of nonlinear programming to the computation of economic equilibria, including the general concept of decentralization. Finally, we show "how" it can be done, outlining the state of the art, and developing a new algorithm that realizes a synthesis between the concepts commonly used in differentiable as well as nondifferentiable optimization. Our approach is as non-technical as possible, and we hope that a nonacquainted reader will be able to follow a non-negligible part of our development. In Section 1, we give the basic concepts underlying nonsmooth optimization and show what it consists of. We also outline the classical methods, which have existed since 1959, aimed at optimizing nondifferentiable problems. In Section 2, we give a list of possible applications, including acceleration of gradient type methods, general decomposition--by prices, by resources, and Benders decomposition--minimax problems, and computation of economic equilibria. In Section 3, we give the most modern methods for nonsmooth optimization, defined around 1975, which were the first general descent methods. In Section 4, we develop a new descent method, which is based on concepts of variable metric, cutting plan approximation and feasible directions. We study its motivation, its convergence, and its flexibility

Nonsmooth Optimization: Use of the Code DYNEPS

One of the aims of the Optimization Task of the System and Decision Sciences Area is to provide computer codes that help to solve certain numerical problems. This paper describes the use of such a code which is being used successfully on a number of IIASA problems, in particular for the Food and Agriculture Program and Human Settlements and Services

Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography

Differentiability of a Support Function of an E-Subgradient

Directional differentiability of a support function of an epsilon-subgradient set-valued mapping is proved and formula for a directional derivative is given

Computing Economic Equilibria through Nonsmooth Optimization

In the first section the problem of nonsmooth optimization is described in general terms, setting the precise hypothesis in mathematical language. Section 2 describes the principles of an example which arises in the context of the linkage of national models of food and agriculture. The general methodology is presented in Section 3, where the algorithm of solution is outlined. Section 4 reports on an extensive set of numerical experiments, both on problems known in the literature, and on the example of Section 2. Finally the paper concludes with some remarks about improvements of the algorithm, which motivate further research on the subject

Some Numerical Experiments with Variable Storage Quasi-Newton Algorithms

This paper relates some numerical experiments with variable storage quasi-Newton methods for the optimization of large-scale models. The basic idea of the recommended algorithm is to start bfgs updates on a diagonal matrix, itself generated by an update formula and adjusted to Rayleigh's ellipsoid of the local Hessian of the objective function in the direction of the change in the gradient. A variational derivation of some rank one and rank two updates in Hilbert spaces is also given

Effect of hypoxia/reoxygenation on the cytokine-induced production of nitric oxide and superoxide anion in cultured osteoarthritic synoviocytes

SummaryObjectiveHypoxia/reoxygenation (H/R) is an important feature in the osteoarthritis (OA) physiopathology. Nitric oxide (NO) is a significant proinflammatory mediator in the inflamed synovium. The purpose of this study was to investigate the effects of H/R on inducible NO synthase (iNOS) activity and expression in OA synoviocytes. In addition we studied the relationship between nitrosative stress and NADPH oxidase (NOX) in such conditions.MethodsHuman cultured synoviocytes from OA patients were treated for 24 h with interleukin 1-β (IL-1β), tumour necrosis factor α (TNF-α) or neither; for the last 6 h, they were submitted to either normoxia or three periods of 1-h of hypoxia followed by 1-h of reoxygenation. NO metabolism (iNOS expression, nitrite and peroxynitrite measurements) was investigated. Furthermore, superoxide anion O2− production, NOX subunit expression and nitrosylation were also assessed.ResultsiNOS expression and nitrite (but not peroxynitrite) production were significantly increased under H/R conditions when compared with to normoxia (P < 0.05). H/R conditions decreased O2− production from ∼0.20 to ∼0.12 nmol min−1 mg proteins−1 (P < 0.05), while NOXs' subunit expression and p47-phox phosphorylation were increased. NOXs and p47-phox were dramatically nitrosylated under H/R conditions (P < 0.05 vs normoxia). Using NOS inhibitors under H/R conditions, p47-phox nitrosylation was prevented and O2− production was restored at normoxic levels (0.21 nmol min−1 mg of proteins−1).ConclusionsOur results provide evidence for an up-regulation of iNOS activity in OA synoviocytes under H/R conditions, associated to a down-regulation of NOX activity through nitrosylation. These findings highlight the importance of radical production to OA pathogenesis, and appraise the metabolic modifications of synovial cells under hypoxia

Thioredoxin 80-Activated-Monocytes (TAMs) Inhibit the Replication of Intracellular Pathogens

BACKGROUND: Thioredoxin 80 (Trx80) is an 80 amino acid natural cleavage product of Trx, produced primarily by monocytes. Trx80 induces differentiation of human monocytes into a novel cell type, named Trx80-activated-monocytes (TAMs). PRINCIPAL FINDINGS: In this investigation we present evidence for a role of TAMs in the control of intracellular bacterial infections. As model pathogens we have chosen Listeria monocytogenes and Brucella abortus which replicate in the cytosol and the endoplasmic reticulum respectively. Our data indicate that TAMs efficiently inhibit intracellular growth of both L. monocytogenes and B. abortus. Further analysis shows that Trx80 activation prevents the escape of GFP-tagged L. monocytogenes into the cytosol, and induces accumulation of the bacteria within the lysosomes. Inhibition of the lysosomal activity by chloroquine treatment resulted in higher replication of bacteria in TAMs compared to that observed in control cells 24 h post-infection, indicating that TAMs kill bacteria by preventing their escape from the endosomal compartments, which progress into a highly degradative phagolysosome. SIGNIFICANCE: Our results show that Trx80 potentiates the bactericidal activities of professional phagocytes, and contributes to the first line of defense against intracellular bacteria

Multicriterion structure/control design for optimal maneuverability and fault tolerance of flexible spacecraft

A multicriterion design problem for optimal maneuverability and fault tolerance of flexible spacecraft is considered. The maneuverability index reflects the time required to perform rest-to-rest attitude maneuvers for a given set of angles, with the postmaneuver spillover within a specified bound. The performance degradation is defined to reflect the maximum possible attitude error after maneuver due to the effect of faults. The fault-tolerant design is to minimize the worst performance degradation from all admissible faults by adjusting the design of the spacecraft. It is assumed that admissible faults can be specified by a vector of real parameters. The multicriterion design for optimal maneuverability and fault tolerance is shown to be well defined, leading to a minimax problem. Analysis for this nonsmooth problem leads to closed-form expressions of the generalized gradient of the performance degradation function with respect to the fault parameters and structural design variables. Necessary and sufficient conditions for the optimum are derived, and the closed-form expressions of the generalized gradients are applied for their interpretation. The bundle method is applicable to this minimax problem. Approximate methods which efficiently solve this minimax problem with relatively little computational difficulties are presented. Numerical examples suggest that it is possible to improve the fault tolerance substantially with relatively little loss in maneuverability.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45242/1/10957_2005_Article_BF02191852.pd