Second-order subdifferential calculus with applications to tilt stability in optimization

The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms

New variational principles with applications to optimization theory and algorithms

In this dissertation we investigate some applications of variational analysis in optimization theory and algorithms. In the first part we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, under the name of tangential extremal principles and rated extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. These developments are in the core geometric theory of variational analysis. Our study includes the basic theory and applications to problems of semi-infinite programming and multiobjective optimization. The second part of this dissertation concerns developing numerical methods of the Newton-type to solve systems of nonlinear equations. We propose and justify a new generalized Newton algorithm based on graphical derivatives. Based on advanced tools of variational analysis and generalized differentiation, we establish the well-posedness and convergence results of the algorithm. Besides, we present a new generalized damped Newton algorithm, which is also known as Newton\u27s method with line-search. Some global convergence results are also justified

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Modelling and Inverse Problems of Control for Distributed Parameter Systems; Proceedings of IFIP(W.G. 7.2)-IIASA Conference, July 24-28, 1989

The techniques of solving inverse problems that arise in the estimation and control of distributed parameter systems in the face of uncertainty as well as the applications of these to mathematical modelling for problems of applied system analysis (environmental issues, technological processes, biomathematical models, mathematical economy and other fields) are among the major topics of research at the Dynamic Systems Project of the Systems and Decision Sciences (SDS) Program at IIASA. In July 1989 the SDS Program was a coorganizer of a regular IFIP (WG 7.2) conference on Modelling and Inverse Problems of Control for Distributed Parameter Systems that was held at IIASA, and was attended by a number of prominent theorists and practitioners. One of the main purpose of this meeting was to review recent developments and perspectives in this field. The proceedings are presented in this volume

The formal theory of pricing and investment for electricity.

The Thesis develops the framework of competitive equilibrium in infinite-dimensional commodity and price spaces, and applies it to the problems of electricity pricing and investment in the generating system. Alternative choices of the spaces are discussed for two different approaches to the price singularities that occur with pointed output peaks. Thermal generation costs are studied first, by using the mathematical methods of convex calculus and majorisation theory, a.k.a. rearrangement theory. Next, the thermal technology, pumped storage and hydroelectric generation are studied by duality methods of linear and convex programming. These are applied to the problems of operation and valuation of plants, and of river flows. For storage and hydro plants, both problems are approached by shadow-pricing the energy stock, and when the given electricity price is a continuous function of time, the plants' capacities, and in the case of hydro also the river flows, are shown to have definite and separate marginal values. These are used to determine the optimum investment. A short-run approach to long-run equilibrium is then developed for pricing a differentiated good such as electricity. As one tool, the Wong-Viner Envelope Theorem is extended to the case of convex but nondifferentiable costs by using the short-run profit function and the profit-imputed values of the fixed inputs, and by using the subdifferential as a multi-valued, generalised derivative. The theorem applies readily to purely thermal electricity generation. But in general the short-run approach builds on solutions to the primal-dual pair of plant operation and valuation problems, and it is this framework that is applied to the case of electricity generated by thermal, hydro and pumped-storage plants. This gives, as part of the long-run equilibrium solution, a sound method of valuing the fixed assets-in this case, the river flows and the sites suitable for reservoirs