Next we discuss some modifications to the sqp method

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Next, we discuss some modifications to the SQP method that aid in solution to the QP subproblem. 7.6.3 The Active Set Strategy The computational cost of solving the QP subproblem can be substantially reduced by only including the active constraints in the subproblem. Accordingly, if the current design point א ȳǡ where Ω denotes the feasible region, then, for some small ߝ ൐ Ͳǡ the set ൌ ൛݅ǣ ݃ ൐ െߝǢ ݅ ൌ ͳǡ ǥ ǡ ݉ൟڂሼ݆ǣ ݆ ൌ ͳǡ ǥ ǡ ݌ሽ denotes the set of potentially active constraints. In the event ב ȳǡ let the current maximum constraint violation be given as: ܸ ሼͲǢ ݃ ǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉Ǣ ห݄ หǡ ݆ ൌ ͳǡ ǥ ǡ ݌ሽ · then, the active constraint set includes: ൌ ൛݅ǣ ݃ ൐ ܸ െ ߝǢ ݅ ൌ ͳǡ ǥ ǡ ݉ൟڂ൛݆ǣ ห݄ ห ൐ ܸ െ ߝǢ ݆ ൌ ͳǡ ǥ ǡ ݌ൟ ² We may note that an inequality constraint at the current design point can be characterized in the following ways: as active ´LI ݃ ൌ Ͳ µ³ as ߝ ¶DFWLYH ´LI ݃ ൐ െߝ µ³ as violated ´ ݂݅ ݃ ൐ Ͳ µ³ or as inactive ´LI ݃ ൑ െߝ µ· whereas, an equality constraint is either active ´ ݄ ൌ Ͳ µ or violated ´ ݄ ് Ͳ µ² The gradients of constraints not in do not need to be computed, however, the numerical algorithm using the potential constraint strategy must be proved to be convergent. Further, from a practical point of view, it is desirable to normalize all constraints with respect to their limit values, so that a uniform ߝ value can be used to check for a constraint condition at the design point.
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Download free eBooks at Fundamental Engineering Optimization Methods 158 ±umerical Optimization Methods Using the active set strategy, the active inequality constraints being known, they can be treated as equality constraints. We, therefore, assume that only equality constraints are present in the active set, and define the QP subproblem as: ݂ ҧ ൌ ࢉ ࢊ ൅ ͳ ʹ 6XEMHFW WR± ࢊ ൌ ࢋ (7.55) Then, using the Lagrangian function approach, the optimality conditions are given as: ࢜ ൅ ࢉ ൅ ࢊ ൌ ૙ǡ ࢊ െ ࢋ ത ൌ ૙ ² They can be simultaneously solved to eliminate the Lagrange multipliers as follows: from the optimality conditions we solve for DV± ࢊ ൌ െࢉ െ ࡺ ࢜ǡ and substitute it in the constraint equation to get: ࢜ ൌ െࡺ ሺࢉ ൅ ࢊሻ ² Next, we substitute back in the optimality condition to get: ࢊ ൌ െሾࡵ െ ࡺ ሺࡺ ିଵ ሿࢉ ൅ ࡺ ሺࡺ ିଵ (7.56) or, more compactly as: ࢊ ൌ ࢊ ൅ ࢊ ³ where in the above expression defines a matrix operator: P = ࡵ െ ࡺ ሺࡺ ିଵ
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