On the reduction of the normality conditions in equality-constrained optimization problems in mechanics |
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Authors: | Danial?Alizadeh Xiao?Qing?Ma Email author" target="_blank">Jorge?AngelesEmail author |
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Institution: | (3) School Math. Sci. Tel-Aviv Univ., Ramat-Aviv, Tel-Aviv, 69978, Israel |
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Abstract: | A large class of problems in mechanics leads to the minimization of an objective function under equality constraints. In fact,
inequality constraints can always be transformed into equality constraints by means of slack variables. The classical approach
to solve equality-constrained problems relies on Lagrange multipliers, whose first-order normality conditions (FONC) lead
to a system of nonlinear algebraic equations. This system of equations involves as many equations as unknowns, composed of
the design variables and Lagrange multipliers, and hence, is amenable to a host of solution methods. In this paper, two methods
to eliminate the Lagrange multipliers are reported, by which a reduced system of normality conditions is obtained. Reduction
is conducted here either symbolically or numerically using an isotropic orthogonal complement
L of the Jacobian matrix of the equality constraints. The relations thus resulting are cast into what is termed the dual form of the FONC. When the problem allows for symbolic calculations, a semi-graphical approach is applied, which leads to the global optimum
of the problem at hand. However, the main novelty of the paper lies in an algorithm that returns the stationary points of
a constrained optimization problem without requiring the closed-form expressions of the dual form of the FONC. Moreover, numerically efficient and stable procedures are given for the intermediate computational steps. The
application of this algorithm is demonstrated with three examples from mechanics. |
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