Worst case asymptotics for some classical optimization problems

In the previous chapters, we presented the fundamental concepts and variants of PSO, as along with a multitude of recent research results. The reported results suggest that PSO can be a very useful tool for solving optimization problems from different scientific and technological fields, especially in cases where classical optimization methods perform poorly or their application involves formidable technical difficulties due to the problem’s special structure or nature. PSO was capable of addressing continuous and integer optimization problems, handling noisy and multiobjective cases, and producing efficient hybrid schemes in combination with specialized techniques or other algorithms in order to detect multiple (local or global) minimizers or control its own parameters.

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A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization

Journal of Mechanical Design ◽

10.1115/1.2202884 ◽

2005 ◽

Vol 128 (4) ◽

pp. 874-883 ◽

Cited By ~ 67

Author(s):

Mian Li ◽

Shapour Azarm ◽

Art Boyars

Keyword(s):

Pareto Front ◽

Optimization Problems ◽

Continuous Functions ◽

Robust Design Optimization ◽

Worst Case ◽

New Approach ◽

Multi Objective ◽

Tolerance Region ◽

Parameter Variations ◽

Gradient Based

We present a deterministic non-gradient based approach that uses robustness measures in multi-objective optimization problems where uncontrollable parameter variations cause variation in the objective and constraint values. The approach is applicable for cases that have discontinuous objective and constraint functions with respect to uncontrollable parameters, and can be used for objective or feasibility robust optimization, or both together. In our approach, the known parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

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Feasibility Robust Optimization via Scenario Generation and Reduction

Volume 2B: 44th Design Automation Conference ◽

10.1115/detc2018-85990 ◽

2018 ◽

Author(s):

Eliot Rudnick-Cohen ◽

Jeffrey W. Herrmann ◽

Shapour Azarm

Keyword(s):

Robust Optimization ◽

Optimization Problems ◽

Computational Cost ◽

Optimization Techniques ◽

Scenario Generation ◽

Test Problems ◽

Optimization Approach ◽

Uncertain Parameters ◽

Worst Case ◽

Robust Solution

Feasibility robust optimization techniques solve optimization problems with uncertain parameters that appear only in their constraint functions. Solving such problems requires finding an optimal solution that is feasible for all realizations of the uncertain parameters. This paper presents a new feasibility robust optimization approach involving uncertain parameters defined on continuous domains without any known probability distributions. The proposed approach integrates a new sampling-based scenario generation scheme with a new scenario reduction approach in order to solve feasibility robust optimization problems. An analysis of the computational cost of the proposed approach was performed to provide worst case bounds on its computational cost. The new proposed approach was applied to three test problems and compared against other scenario-based robust optimization approaches. A test was conducted on one of the test problems to demonstrate that the computational cost of the proposed approach does not significantly increase as additional uncertain parameters are introduced. The results show that the proposed approach converges to a robust solution faster than conventional robust optimization approaches that discretize the uncertain parameters.

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Grand Tour Algorithm: Novel Swarm-Based Optimization for High-Dimensional Problems

Processes ◽

10.3390/pr8080980 ◽

2020 ◽

Vol 8 (8) ◽

pp. 980

Author(s):

Gustavo Meirelles ◽

Bruno Brentan ◽

Joaquín Izquierdo ◽

Edevar Luvizotto

Keyword(s):

Systems Engineering ◽

Collective Behavior ◽

Optimization Problems ◽

Optimization Technique ◽

Optimal Solution ◽

Sensitivity Analyses ◽

Grand Tour ◽

Agent Based ◽

Local Optimal Solution ◽

Classical Optimization

Agent-based algorithms, based on the collective behavior of natural social groups, exploit innate swarm intelligence to produce metaheuristic methodologies to explore optimal solutions for diverse processes in systems engineering and other sciences. Especially for complex problems, the processing time, and the chance to achieve a local optimal solution, are drawbacks of these algorithms, and to date, none has proved its superiority. In this paper, an improved swarm optimization technique, named Grand Tour Algorithm (GTA), based on the behavior of a peloton of cyclists, which embodies relevant physical concepts, is introduced and applied to fourteen benchmarking optimization problems to evaluate its performance in comparison to four other popular classical optimization metaheuristic algorithms. These problems are tackled initially, for comparison purposes, with 1000 variables. Then, they are confronted with up to 20,000 variables, a really large number, inspired in the human genome. The obtained results show that GTA clearly outperforms the other algorithms. To strengthen GTA’s value, various sensitivity analyses are performed to verify the minimal influence of the initial parameters on efficiency. It is demonstrated that the GTA fulfils the fundamental requirements of an optimization algorithm such as ease of implementation, speed of convergence, and reliability. Since optimization permeates modeling and simulation, we finally propose that GTA will be appealing for the agent-based community, and of great help for a wide variety of agent-based applications.

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An interior point algorithm for solving linear optimization problems using a new trigonometric kernel function

Filomat ◽

10.2298/fil2005471f ◽

2020 ◽

Vol 34 (5) ◽

pp. 1471-1486

Author(s):

S. Fathi-Hafshejani ◽

Reza Peyghami

Keyword(s):

Kernel Function ◽

Interior Point ◽

Optimization Problems ◽

Linear Optimization ◽

Interior Point Algorithm ◽

Worst Case ◽

Complexity Bounds ◽

Linear Optimization Problems ◽

Primal Dual ◽

The Given

In this paper, a primal-dual interior point algorithm for solving linear optimization problems based on a new kernel function with a trigonometric barrier term which is not only used for determining the search directions but also for measuring the distance between the given iterate and the ?-center for the algorithm is proposed. Using some simple analysis tools and prove that our algorithm based on the new proposed trigonometric kernel function meets O (?n log n log n/?) and O (?n log n/?) as the worst case complexity bounds for large and small-update methods. Finally, some numerical results of performing our algorithm are presented.

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A Dynamical Alternating Direction Multiplier Method for Two-Block Optimization Problems

10.21203/rs.3.rs-718554/v1 ◽

2021 ◽

Author(s):

Miantao Chao ◽

Liqun Liu

Keyword(s):

Dynamical System ◽

Optimization Problems ◽

Suitable Condition ◽

Time Discretization ◽

Alternating Direction Method ◽

Lagrangian Function ◽

Method Of Multipliers ◽

Worst Case ◽

Alternating Direction ◽

Separable Optimization

Abstract In this paper, we propose a dynamic alternating direction method of multipliers for two-block separable optimization problems. The well-known classical ADMM can be obtained after the time discretization of the dynamical system. Under suitable condition, we prove that the trajectory asymptotically converges to a saddle point of the Lagrangian function of the problems. When the coefficient matrices in the constraint are identiy matrices, we prove the worst-case O(1/t) convergence rate in ergodic sense.

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