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Minimization Problem

Minimization Problem. vs. Kuhn-Tucker Stationary-point Problem. such that. MP:. KTSP:. Find. such that. Lagrangian Function. Let. and. are convex then. If. is convex. For a fixed. , if. then. Above result is a sufficient condition if. is convex. KTSP with Equality Constraints?.

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Minimization Problem

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  1. Minimization Problem vs. Kuhn-Tucker Stationary-point Problem such that MP: KTSP: Find such that

  2. Lagrangian Function Let and are convex then • If is convex. • For a fixed , if then • Above result is a sufficient condition if is convex.

  3. KTSP with Equality Constraints? are linear functions) (Assume such that KTSP: Find

  4. KTSP with Equality Constraints Find such that KTSP: • Let and then is free variable

  5. Generalized Lagrangian Function Let and are convex and is linear then • If is convex. • For fixed , if then • Above result is a sufficient condition if is convex.

  6. Lagrangian Dual Problem subject to

  7. Lagrangian Dual Problem subject to subject to where

  8. Linear Programs: Primal vs. Dual

  9. Lagrangian Function of Primal LP

  10. Weak Duality Theorem be a feasible solution of the primal Let problem and a feasible solution of the dualproblem. Then Corollary:

  11. Weak Duality Theorem where Corollary: If , then and and solve the primal and dual problem respectively. In this case,

  12. Saddle Point of Lagrangian satisfying Let is called Then The saddle point of the Lagrangian function

  13. Saddle point of

  14. Dual Problem of Linear Program Primal LP subject to Dual LP subject to • All duality theorems hold andwork perfectly!

  15. If is primal feasible and is dual feasible, then the dual objective function evaluated at is less than or equal to the primal objective function evaluated at Weak Duality Theorem in LP

  16. Strong Duality Theorem in LP Exactly one of the following three alternatives holds: • Both primal and dual problems are feasible and consequently both have optimal solutions with equal extrema. • Exactly one of the problems is infeasible and consequently the other problem has an unbounded objective function on its feasible region. • Both primal and dual problems are infeasible.

  17. solve (P) and (D) iff satisfy (KKT) KKT Conditions of Linear Program (KKT)

  18. Application of LP Duality (I) Farkas’Lemma For any matrix and any vector either or but never both.

  19. Application of LP Duality (II) LSQ-Normal Equation Always Has a Solution For any matrix and any vector consider always has a solution. Claim:

  20. The Most Important Concepts 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 KKT conditions • There might exist decent direction but move along this direction will leave out the feasible region

  21. Minimization Problem vs. Kuhn-Tucker Stationary-point Problem such that MP: KTSP: Find such that

  22. Gordan’s Theorem (Gordan 73) For any given matrix either or but NEVER both

  23. subject to Dual Problem of Strictly Convex Quadratic Program Primal QP subject to Withstrictly convexassumption, we have Dual QP

  24. Weak Duality Theorem be a feasible solution of the primal Let problem and a feasible solution of the dualproblem. Then Corollary:

  25. Weak Duality Theorem where Corollary: If , then and and solve the primal and dual problem respectively. In this case,

  26. satisfying Let is called Then • The triple is a saddle point of Lagarangian for the primal problem iff solves (P) and solves (D) and there is no duality gap Saddle Point of Lagrangian The saddle point of the Lagrangian function

  27. If is a local solution for the QP then such that Conversely, if satisfy KKT conditions and is psd, then is a global solution KKT Conditions of Quadratic Program Primal QP subject to

  28. Quadratic Programming Solver in MATLAB X=QUADPROG(H,f,A,b) attempts to solve the quadratic programming problem: min 0.5*x'*H*x + f'*x subject to: A*x <= b x X=QUADPROG(H,f,A,b,Aeq,beq) solves the problem above while additionally satisfying the equality constraints Aeq*x = beq. X=QUADPROG(H,f,A,b,Aeq,beq,LB,UB) defines a set of lower and upper bounds on the design variables, X, so that the solution is in the range LB <= X <= UB. Use empty matrices for LB and UB if no bounds exist. Set LB(i) = -Inf if X(i) is unbounded below; set UB(i) = Inf if X(i) is unbounded above.

  29. Quadratic Programming Solver in MATLAB X=QUADPROG(H,f,A,b,Aeq,beq,LB,UB,X0) sets the starting point to X0. You can type “help quadprog” in MATLAB to get more information!

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