Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

PROBLEM: problem_linear
minimizing linear
subject to
b1 ≤Ax ≤b2 (multiple linear constraints)
Box constraints (types of variables)
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Linear = Linear function
Box constraints = constraints on individual decision variables
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Download other datasets in Run-File Environment.
Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.83GHz (sec)
Dataset2	Problem statement	Data	Solution	158	N/A	0.000218	6.56
Dataset3	Problem statement	Data	Solution	158	N/A	0.000015	9.41
Dataset4	Problem statement	Data	Solution	158	N/A	0.000006	13.11
Dataset5	Problem statement	Data	Solution	158	N/A	0.000003	12.93
Dataset6	Problem statement	Data	Solution	158	N/A	0.00001	8.37
Dataset7	Problem statement	Data	Solution	158	N/A	1.0269e-06	7.43
Dataset8	Problem statement	Data	Solution	158	N/A	0.000006	16.32
Dataset9	Problem statement	Data	Solution	158	N/A	0.000013	13.5
Dataset10	Problem statement	Data	Solution	158	N/A	1.3339e-05	17.35
Dataset11	Problem statement	Data	Solution	158	N/A	1.4316e-05	19.64
Dataset12	Problem statement	Data	Solution	158	N/A	1.6291e-05	28.29
Dataset13	Problem statement	Data	Solution	158	N/A	0.00002	2.83
Dataset14	Problem statement	Data	Solution	158	N/A	0.000026	18.04
Dataset15	Problem statement	Data	Solution	158	N/A	0.000037	8.11
Dataset16	Problem statement	Data	Solution	158	N/A	0.000053	10.31
Dataset17	Problem statement	Data	Solution	158	N/A	0.000079	14.77

CASE STUDY SUMMARY
This Case Study demonstrates optimization of parameters of hinged beam under the influence of a number of periodic concentrated forces for excitation and formation of wave motion. The use of mathematical methods that lead to optimization problems is the traditional problems in the design of mechanical devices with optimum characteristics specified criterial (Balandin (1995), Komkov (1972)). However, in some cases, an optimization approach leads to mixed multi-extremal problems of constrained minimization of functionals, which are difficult to solve (Banichuk (1990), Banichuk, and Klimov (1991)). In this case study we consider a model problem of the vibrations of hinged beam under the influence of a number of periodic concentrated forces. The problem is to choose the parameters of force influence (amount of forces, their amplitudes, phases, and points of their application) to install the vibrations of a beam well (according to some criterion) met the specified parameters waveform.
The Case Study solves the problem for a fixed frequency (k = 2.5) with a serial relaxing constraints imposed on the amount of forces (I = 2, 3, 5, 6, 7, 8, 9, 10). Besides we solve the problem to determine the optimal characteristics for 5 forces (I = 5) within a frequency range comprising the resonance (k = 2.1, 2.3, 2.5, 2.7, 2.9, 3.1, 3.3, 3.5, 3.7).

References
• Balandin D. (1995), “On the optimal vibration absorption in elastic objects”. Applied Mathematics and Mechanics, Vol. 59, # З (In Russian).
• Komkov V. (1972), Optimal control theory for the damping of vibrations of simple elastic systems. Springer Verlag. Berlin. Heidelberg, New York.
• Banichuk N.V. (1990), Introduction to Optimization of Structures, Springer-Verlag, New York.
• Banichuk N.V., and D.M.Klimov (1991), Dynamical Problems of Rigid-Elastic Systems and Structures, Springer-Verlag, , New York.