fundamental-engineering-optimization-methods.pdf

Once a search direction d has been determined a step

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Once a search direction d has been determined, a step-size along d needs to be computed by solving the line search problem. We next discuss the descent function approach that is used to resolve the line search step in the SQP solution process. 7.6.1 Descent Function Approach In SQP methods, the line search solution is based on minimization of a descent function that penalizes constraint violations. The following descent function has been proposed in literature (Arora, p. 521): Ȱሺ࢞ሻ ൌ ݂ሺ࢞ሻ ൅ ܴܸሺ࢞ሻ (7.50)
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 155 ±umerical Optimization Methods where ݂ሺ࢞ሻ represents the cost function value, ܸሺ࢞ሻ represents the maximum constraint violation, and ܴ ൐ Ͳ is a penalty parameter. The descent function value at the current iteration is expressed as: Ȱ ൌ ݂ ൅ ܴܸ ³ ܴ ൌ ሼܴ ǡ ݎ (7.51) where ܴ is the current value of the penalty parameter, ݎ is the current sum of the Lagrange multipliers, and ܸ is the maximum constraint violation in the current step. The latter parameters are computed as: ݎ ൌ σ ݑ ௜ୀଵ ൅ σ หݒ ௝ୀଵ ܸ ሼͲǢ ݃ ǡ ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ݉Ǣ ห݄ หǡ ݆ ൌ ͳǡ ǥ ǡ ݌ሽ (7.52) where absolute values of the Lagrange multipliers and constraint violations for equality constraints are used. Next, the line search subproblem is defined as: Ȱሺߙሻ ൌ Ȱ൫࢞ ൅ ߙࢊ (7.53) The above problem may be solved via the line search methods described in Sec. 7.2. An algorithm for solving the SQP problem is presented below: SQP Algorithm (Arora, p. 526) : Initialize: choose ǡ ܴ ൌ ͳǡ ߝ ൐ Ͳǡ ߝ ൐ Ͳ ² For ݇ ൌ Ͳǡͳǡʹǡ ǥ 1. Compute ݂ ǡ ݃ ǡ ݄ ǡ ࢉǡ ܾ ǡ ݁ · FRPSXWH ܸ ² 1. Formulate and solve the QP subproblem to obtain d k and the Lagrange multipliers DQG ² 2. If ܸ ൑ ߝ DQG ฮࢊ ฮ ൑ ߝ ³ VWRS² 3. Compute ܴ · formulate and solve line search subproblem to obtain ߙ 4. Set ௞ାଵ ՚ ࢞ ൅ ߙࢊ ǡ ܴ ௞ାଵ ՚ ܴǡ ݇ ՚ ݇ ൅ ͳ ² It can be shown that the above algorithm is convergent, i.e., Ȱ൫࢞ ൯ ൑ Ȱሺ࢞ ³ DQG WKDW converges to the KKT point in the case of general constrained optimization problems (Arora, p. 525).
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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 156 ±umerical Optimization Methods 7.6.2 SQP with Approximate Line Search The above SQP algorithm can be used with approximate line search methods, similar to Arjimo’s rule (Sec. 7.2.2) as follows: let ݐ ǡ ݆ ൌ Ͳǡͳǡ ǥ denote a trial step size, ௞ାଵǡ௝ denote the trial design point, ݂ ௞ାଵǡ௝ ൌ ݂ሺ ࢞ ௞ାଵǡ௝ denote the function value at the trial solution, and Ȱ ௞ାଵǡ௝ ൌ ݂ ௞ାଵǡ௝ ൅ ܴܸ ௞ାଵǡ௝ denote the penalty function at the trial solution. The trial solution is required to satisfy the following
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