The Study Of Composite Proximal Bundle Method And Its Dual Problem

The terminology of nonsmooth optimization is firstly proposed by Bailinski and Wolfein1975．This kind of problems stems from numerous applications area，such as economics，mechanics，engineering and optimal control．Bundle method is currently considered to be oneof the most effective and the most promising methods to solve this kind of problems，theoriginal intention of proposing bundle method is to solve a class of extremum problems．Thestudy of bundle methods is related with convex analysis，linear programming，nonlinearprogramming，non-smooth analysis and other related areas．For solving the nonsmooth optimization problems，we have many different kinds ofmethods．In fact, the methods can be divided into two major types，namely subgradientmethod and bundle method. Both methods are based on the assumption of the existence of theunique objective function value and subgradient value at each point. Subgradient methodbased on subgradient is the method of solving nonsmooth optimization problems，which iseasier to implement，but it may lead to the failure in its convergence，optimum detection andsubgradient approximation．Cutting-planes method is the ideological foundation of bundlemethod，and it makes full use of the information got from the black box to constructpiecewise affine model of the objective function in the original problem．On the basis ofcutting-planes method，the general bundle method is introduced．This method is widely usedat present，and it can retain the related information of iteration point generated by the previousiteration and subgradient．It establishes a bundle of information．Therefore, we can producethe next iteration point by using the previous established information at the current iterationstep．With the progress of the iteration, the bundle of information will be bigger and bigger，which brings the problems of storage and calculation．For this，special aggregation appears inbundle method．This paper focuses on the study of composite proximal bundle method for compositeform objective function．The thesis is organized as follows：In the first chapter，firstly，the historical background and the research status of thenonsmooth optimization problems is introduced．The appearance of nonsmooth functionsbrings many difficulties for solving the optimization problems．To overcome the difficulties in calculation and theoretical analysis， many different methods emerge as the timesrequire．Then this paper separately introduces the basic thought of cutting-planes method andgeneral bundle method，which lays a theoretical foundation for the further study on the secondand the third chapter．In the second chapter，the basic thought of nonsmooth composite bundle method isgiven，the conceptual model of the objective function is introduce．The cutting-planes modelconstruction of conceptual model differs from the cutting-planes model construction ofobjective function in general bundle method．In this chapter，we covert the original problemswith the composite objective function of positively homogeneous convex function andsmooth mapping to a series of quadratic programming problems by employing penalizedthought of general bundle method．For the stable sub-problems，we study it by adopting dualspace theory，discuss and prove related properties in its dual space，depicts the relationshipbetween the original problem and dual problem. Furthermore， we define aggregatelinearization，and summarize some properties of aggregate linearization．Meanwhile, the basicframework of the corresponding algorithm is presented in detail．In the third chapter，we construct a special composite objective function，and the outerfunction is positively homogeneous convex function， the inner function is linearfunction．Then we get the cutting-planes model of composite objective function，and use thisspecial cutting-planes model to construct the stable penalized sub-problem corresponding tothe original problems．As well，we adopt dual space theory to find the dual problem of thispenalized sub-problem，and carry out the research of various properties．At the same time，we introduce the basic steps of the algorithm，carry out in detail the convergence analysis．Forthe convergence analysis，we mainly discuss two cases．When the algorithm produces the lastdescent step，followed by infinitely many null steps，the sequence of candidate pointsconverges to the minimum point of the original problem under the condition that theparameter kis an increasing sequence and has the upper and lower bounds. When algorithmgenerates an infinite number of descent-steps，we also obtain the corresponding convergenceresult by introducing the concept of minimizing sequence．...