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Published byDuane Turner Modified over 9 years ago
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The Most Important Concept in Optimization (minimization) A point is said to be an optimal solution of a unconstrained minimization if there exists no decent direction A point is said to be an optimal solution of a constrained minimization if there exists no feasible decent direction There might exist decent direction but move along this direction will leave out the feasible region
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Decent Direction of Move alone the decent direction for a certain stepsize will decrease the objective function value i.e.,
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Feasible Direction of Move alone the feasible direction from for a certain stepsize will not leave the feasible region i.e., where is the feasible region.
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Steep Decent with Exact Line Search Start with any. Having stop if (i) Steep Decent Direction : (ii) Finding Stepsize by Exact Line Search :
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Minimum Principle Let be a convex and differentiable function be the feasible region. Example:
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Minimization Problem vs. Kuhn-Tucker Stationary-point Problem Findsuch that KTSP: MP: such that
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Lagrangian Function For a fixed Let and If are convex then is convex., if then Above result is a sufficient condition if is convex.
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KTSP with Equality Constraints? (Assume are linear functions) KTSP: Find such that
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KTSP with Equality Constraints KTSP: Findsuch that Ifandthen is free variable
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Generalized Lagrangian Function For fixed, if then Letand If are convex andis linear then is convex. Above result is a sufficient condition if is convex.
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Lagrangian Dual Problem subject to
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Lagrangian Dual Problem subject to where
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Weak Duality Theorem Let be a feasible solution of the primal problem and a feasible solution of the dual problem. Then Corollary:
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Weak Duality Theorem Corollary:If where and, thenand solve the primal and dual problem respectively. In this case,
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Saddle Point of Lagrangian Let satisfying Then is called The saddle point of the Lagrangian function
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Dual Problem of Linear Program subject to Primal LP Dual LP subject to ※ All duality theorems hold and work perfectly!
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Application of LP Duality (I) Farkas ’ Lemma For any matrixand any vector either or but never both.
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Application of LP Duality (II) LSQ-Normal Equation Always Has a Solution For any matrixand any vector consider Claim: always has a solution.
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Dual Problem of Strictly Convex Quadratic Program subject to Primal QP With strictly convex assumption, we have Dual QP subject to
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