/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka---?ojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward---backward algorithms with semi-algebraic problem's data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.

/pdf/proximal-alternating-linearized-minimization-for-nonconvex-lbqbrhi298.pdf

Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Nonlinear Programming: A Unified Approach

This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

/pdf/engineering-and-economic-applications-of-complementarity-1nz165yv0c.pdf

Engineering and Economic Applications of Complementarity Problems

This book is about the theoretical foundations of optimization algorithms, and also provides practical insights on how such methods should be implemented and applied. The goal of this text is not focused on in-depth details of state-of-the-art methods, but rather to provide a good foundation of the tried and trusted methods and how they can be used to solve real-world problems efficiently. The target audience for the text seems to be graduate students and engineers in industry as a good reference or self-study. However, it does not lend itself to easy adaption in a classroom setting due to the fact that only some chapters have problem sets and most chapters do not have a summary section. The book has four main parts each written by the one of the different authors. Two of the parts deal with standard unconstrained and constrained optimization techniques. The other two section deal with more complicated issues including solving problems that exhibit nonsmooth behavior and interior point methods. Solving optimization problems that have nonsmooth objectives or constraints is a formidable challenge because sensitivity information cannot be calculated using traditional approaches; the book provides a variety of ways to approach such problems. Interior point methods have been gaining popularity in the engineering optimization community for their efficiency and ability to maintain feasibility throughout the optimization process. There have been many textbooks written on the subject of numerical optimization. This book has the same flavor as many of these past books. A book with a very similar title by Nocedal and Wright [1] covered much of the same material, but it did not address nonsmooth problems. A classical book in the field by Flecther [2] considers nonsmooth problems but not interior point methods. Also, if one is more interested in comparing the performance of the various methods applied to engineering design problems, the book by Vanderplaats [3] may be more suitable. The appeal of this book is in the authors’ attention to detail and their advice on practical implementation issues. For many of the different optimization algorithms detailed flow charts are given. They also provide adequate examples to help the reader understand the methods better and explore possible pitfalls. There are, however, a few downsides of this book. The main one is that through translation of the book to English, it is evident that some of the finer points were lost and in some cases the text is hard to understand. The book could also have benefited from more figures; there are only 26. Finally, each of the dif

Numerical Optimization: Theoretical and Practical Aspects

Convex Analysis and Set Valued Maps: A Review.- Asymptotic Cones and Functions.- Existence and Stability in Optimization Problems.- Minimizing and Stationary Sequences.- Duality in Optimization Problems.- Maximal Monotone Maps and Variational Inequalities.

Asymptotic cones and functions in optimization and variational inequalities

Interior gradient (subgradient) and proximal methods for convex constrained minimization have been much studied, in particular for optimization problems over the nonnegative octant. These methods are using non-Euclidean projections and proximal distance functions to exploit the geometry of the constraints. In this paper, we identify a simple mechanism that allows us to derive global convergence results of the produced iterates as well as improved global rates of convergence estimates for a wide class of such methods, and with more general convex constraints. Our results are illustrated with many applications and examples, including some new explicit and simple algorithms for conic optimization problems. In particular, we derive a class of interior gradient algorithms which exhibits an $O(k^{-2})$ global convergence rate estimate.

Interior Gradient and Proximal Methods for Convex and Conic Optimization

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.

https://link.springer.com/content/pdf/10.1023/A:1008607511915.pdf

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities. Developing a simple Lagrangian duality scheme which is combined with the recent logarithmic-quadratic proximal (LQP) theory introduced by the authors, we derive three algorithms for solving the variational inequality (VI) problem. This provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms. We prove primal, dual, and primal-dual convergence under very mild assumptions, eliminating all the usual assumptions used until now in the literature for related algorithms. Applications to complementarity problems are also discussed.

Lagrangian Duality and Related Multiplier Methods for Variational Inequality Problems

We consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature. We provide a systematic way to generate penalty and barrier functions in this class, and we analyze the existence of primal and dual optimal paths generated by these penalty methods, as well as their convergence to the primal and dual optimal sets. For linear programming we prove that these optimal paths converge to single points.

Alfred Auslender

Papers

Asymptotic cones and functions in optimization and variational inequalities

Interior Gradient and Proximal Methods for Convex and Conic Optimization

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

Lagrangian Duality and Related Multiplier Methods for Variational Inequality Problems

Asymptotic analysis for penalty and barrier methods in convex and linear programming