Transcription of Constrained Optimization Using Lagrange Multipliers
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Constrained Optimization Using Lagrange Multipliers
Constrained Optimization Using Lagrange MultipliersCEE 201L. Uncertainty, Design, and OptimizationDepartment of Civil and Environmental EngineeringDuke UniversityHenri P. Gavin and Jeffrey T. ScruggsSpring 2020In optimal design problems, values for a set ofndesign variables, (x1,x2, xn), areto be found that minimize a scalar-valued objective function of the design variables, suchthat a set ofminequality constraints, are satisfied. Constrained Optimization problems aregenerally expressed asminx1,x2, ,xnJ=f(x1,x2, ,xn)such thatg1(x1,x2, ,xn) 0g2(x1,x2, ,xn) (x1,x2, ,xn) 0(1)If the objective function is quadratic in the design variables and the constraint equations arelinearly independent, the Optimization problem has a unique the simplest Constrained minimization problem:minx12kx2wherek >0such thatx b .(2)This problem has a single design variable, the objective function is quadratic (J=12kx2),there is a single constraint inequality, and it is linear inx(g(x) =b x).
Jul 10, 2020 · If the objective function is quadratic in the design variables and the constraint equations are linearly independent, the optimization problem has a unique solution. Consider the simplest constrained minimization problem: min x 1 2 kx2 where k>0 such that x≥b. (2) This problem has a single design variable, the objective function is quadratic ...
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