291 research outputs found
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
A DC Programming Approach for Solving Multicast Network Design Problems via the Nesterov Smoothing Technique
This paper continues our effort initiated in [9] to study Multicast
Communication Networks, modeled as bilevel hierarchical clustering problems, by
using mathematical optimization techniques. Given a finite number of nodes, we
consider two different models of multicast networks by identifying a certain
number of nodes as cluster centers, and at the same time, locating a particular
node that serves as a total center so as to minimize the total transportation
cost through the network. The fact that the cluster centers and the total
center have to be among the given nodes makes this problem a discrete
optimization problem. Our approach is to reformulate the discrete problem as a
continuous one and to apply Nesterov smoothing approximation technique on the
Minkowski gauges that are used as distance measures. This approach enables us
to propose two implementable DCA-based algorithms for solving the problems.
Numerical results and practical applications are provided to illustrate our
approach
Fuzzy Bilevel Optimization
In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems.
In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1
1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2
1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Preliminaries 11
2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11
2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
3 Optimization problem with fuzzy objective 19
3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25
4 Linear optimization with fuzzy objective 27
4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36
4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
5 Optimality conditions 47
5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48
5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49
5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51
5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52
5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54
5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Fuzzy linear optimization problem over fuzzy polytope 59
6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Bilevel optimization with fuzzy objectives 73
7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78
7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87
8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9 Conclusions 95
Bibliography 9
Economic inexact restoration for derivative-free expensive function minimization and applications
The Inexact Restoration approach has proved to be an adequate tool for
handling the problem of minimizing an expensive function within an arbitrary
feasible set by using different degrees of precision in the objective function.
The Inexact Restoration framework allows one to obtain suitable convergence and
complexity results for an approach that rationally combines low- and
high-precision evaluations. In the present research, it is recognized that many
problems with expensive objective functions are nonsmooth and, sometimes, even
discontinuous. Having this in mind, the Inexact Restoration approach is
extended to the nonsmooth or discontinuous case. Although optimization phases
that rely on smoothness cannot be used in this case, basic convergence and
complexity results are recovered. A derivative-free optimization phase is
defined and the subproblems that arise at this phase are solved using a
regularization approach that take advantage of different notions of
stationarity. The new methodology is applied to the problem of reproducing a
controlled experiment that mimics the failure of a dam
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