Nonsmooth and derivative-free optimization based hybrid methods and applications

"In this thesis, we develop hybrid methods for solving global and in particular, nonsmooth optimization problems. Hybrid methods are becoming more popular in global optimization since they allow to apply powerful smooth optimization techniques to solve global optimization problems. Such methods are able to efficiently solve global optimization problems with large number of variables. To date global search algorithms have been mainly applied to improve global search properties of the local search methods (including smooth optimization algorithms). In this thesis we apply rather different strategy to design hybrid methods. We use local search algorithms to improve the efficiency of global search methods. The thesis consists of two parts. In the first part we describe hybrid algorithms and in the second part we consider their various applications." -- taken from Abstract.Operational Research and Cybernetic

A nonsmooth exclusion test for finding all solutions of nonlinear equations

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.Includes bibliographical references (p. 93-94).A new approach is proposed for finding all solutions of systems of nonlinear equations with bound constraints. The zero finding problem is converted to a global optimization problem whose global minima with zero objective value, if any, correspond to all solutions of the initial problem. A branch-and-bound algorithm is used with McCormick's nonsmooth convex relaxations to generate lower bounds. An inclusion relation between the solution set of the relaxed problem and that of the original non-convex problem is established which motivates a method to generate automatically reasonably close starting points for a local Newton-type method. A damped-Newton method with natural level functions employing the restrictive monotonicity test is employed to find solutions robustly and rapidly. Due to the special structure of the objective function, the solution of the convex lower bounding problem yields a nonsmooth root exclusion test which is found to perform better than earlier interval based exclusion tests. The Krawczyk operator based root inclusion and exclusion tests are also embedded in the proposed algorithm to refine the variable bounds for efficient fathoming of the search space. The performance of the algorithm on a variety of test problems from the literature is presented and for most of them the first solution is found at the first iteration of the algorithm due to the good starting point generation.by Vinay Kumar.S.M

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

Hyperbolic smoothing in nonsmooth optimization and applications

Nonsmooth nonconvex optimization problems arise in many applications including economics, business and data mining. In these applications objective functions are not necessarily differentiable or convex. Many algorithms have been proposed over the past three decades to solve such problems. In spite of the significant growth in this field, the development of efficient algorithms for solving this kind of problem is still a challenging task. The subgradient method is one of the simplest methods developed for solving these problems. Its convergence was proved only for convex objective functions. This method does not involve any subproblems, neither for finding search directions nor for computation of step lengths, which are fixed ahead of time. Bundle methods and their various modifications are among the most efficient methods for solving nonsmooth optimization problems. These methods involve a quadratic programming subproblem to find search directions. The size of the subproblem may increase significantly with the number of variables, which makes the bundle-type methods unsuitable for large scale nonsmooth optimization problems. The implementation of bundle-type methods, which require the use of the quadratic programming solvers, is not as easy as the implementation of the subgradient methods. Therefore it is beneficial to develop algorithms for nonsmooth nonconvex optimization which are easy to implement and more efficient than the subgradient methods. In this thesis, we develop two new algorithms for solving nonsmooth nonconvex optimization problems based on the use of the hyperbolic smoothing technique and apply them to solve the pumping cost minimization problem in water distribution. Both algorithms use smoothing techniques. The first algorithm is designed for solving finite minimax problems. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study the main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in GAMS. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem. The second nonsmooth optimization algorithm we developed was used to demonstrate how smooth optimization methods can be applied to solve general nonsmooth (nonconvex) optimization problems. In order to do so we compute subgradients from some neighborhood of the current point and define a system of linear inequalities using these subgradients. Search directions are computed by solving this system. This system is solved by reducing it to the minimization of the convex piecewise linear function over the unit ball. Then the hyperbolic smoothing function is applied to approximate this minimization problem by a sequence of smooth problems which are solved by smooth optimization methods. Such an approach allows one to apply powerful smooth optimization algorithms for solving nonsmooth optimization problems and extend smoothing techniques for solving general nonsmooth nonconvex optimization problems. The convergence of the algorithm based on this approach is studied. The proposed algorithm was implemented in Fortran 95. Preliminary results of numerical experiments are reported and the proposed algorithm is compared with an other five nonsmooth optimization algorithms. We also implement the algorithm in GAMS and compare it with GAMS solvers using results of numerical experiments.Doctor of Philosoph