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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The Fundamental Theorem of Linear Programming says that if there is a feasible solution, there is a basic feasible solution, and if there is an optimal feasible solution, there is an optimal basic feasible solution. The linear programming problem is thus reduced to searching among the set of basic solutions for an optimal solution. This set is, of course, finite, containing as many as n!/[m!(n-m)!] points. In practice, this will be a very large number, making it imperative that one use some efficient search procedure in seeking an optimal solution. The most important of such procedures is the simplex method, details of which may be found in the references.
Optimization Methods
Published in William F. Ames, George Cain, Y.L. Tong, W. Glenn Steele, Hugh W. Coleman, Richard L. Kautz, Dan M. Frangopol, Paul Norton, Mathematics for Mechanical Engineers, 2022
The Fundamental Theorem of Linear Programming says that if there is a feasible solution, there is a basic feasible solution, and if there is an optimal feasible solution, there is an optimal basic feasible solution. The linear programming problem is thus reduced to searching among the set of basic solutions for an optimal solution. This set is, of course, finite, containing as many as n!/[m!(n − m)!] points. In practice, this will be a very large number, making it imperative that one use some efficient search procedure in seeking an optimal solution. The most important of such procedures is the simplex method, details of which may be found in the references.
A linear programming approach for designing multilevel PWM waveforms
Published in International Journal of Control, 2021
Shravan Mohan, Bharath Bhikkaji, C Poongothai, Krishna Vasudevan
By substituting and using the fact that for matrices U, X, V, W of appropriate dimensions , the LP in (12) is equivalent to: where Now, the fundamental theorem of linear programming, Boyd and Vandenberghe (2004) states that a solution (if it exists) to the standard linear program (with constraints forming a bounded polytope): is a basic feasible solution. Moreover, when the number of constraints are lesser than the number of variables , a basic feasible solution vector has at most non-zero elements. In (13), the analogue of the matrix Q is . Its rank is bounded by: Therefore, a basic feasible solution to (13) would have at least zero elements. Suppose that the number of zero elements in the row of is given by . Then Since is stochastic (each row adds to 1), Hence, (17) and (18) imply that at least of 's have to be equal to . That is, has at least rows with zero elements. In other words, must have at least elements in L.