MIE376 Lec 5 - NLPQPv2

MIE376 Lec 5 - NLPQPv2 - Non-linear Optimization...

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MIE376 - Mathematical Programming 1 Non-linear Optimization Introduction Unconstrained Optimization Constrained Optimization – Equality Constraints Constrained Optimization – Inequality Constraints (NLP) Application of KKT to Quadratic Programming (QP) Application of KKT to Bi-Level LP (BLLP) Remaining Topics and Return to Excel
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MIE376 - Mathematical Programming 2 Non-linear Optimization Introduction Unconstrained Optimization Constrained Optimization – Equality Constraints Constrained Optimization – Inequality Constraints (NLP) Quadratic Programming (QP) Remaining Topics Return to Excel
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MIE376 - Mathematical Programming 3 Introduction ± OR problems usually ² deal with man-made systems ² often ignore uncertainty ² involve linear and integer domains ² Æ LP and IP are all we need ± Sometimes ² Involve elements from the natural world ² Other sources of non-linearity ² Require non-linear optimization
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MIE376 - Mathematical Programming 4 Convex Functions and Sets ± a function is convex if and only if for any two vector x and y and any value of a scalar λ in the range 0 to 1: f( λ x + (1- λ )y) ≤λ f( x ) + (1- λ ) f( y ) ± Constraints form a convex set if and only if for any two vectors, x and y satisfying the constraints, then for any value of a scalar λ in the range 0 to 1, the vector λ a + (1- λ )b also satisfies the constraints ± Except for one case we will always assume all functions and constraints are Continuous, Differentiable and Convex.
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MIE376 - Mathematical Programming 5 Overview
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MIE376 - Mathematical Programming 6 Non-linear Optimization Introduction Unconstrained Optimization Constrained Optimization – Equality Constraints Constrained Optimization – Inequality Constraints (NLP) Application of KKT to Quadratic Programming (QP) Application of KKT to Bi-Level LP (BLLP) Remaining Topics and Return to Excel
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MIE376 - Mathematical Programming 7 Some types of objectives… ± Separable ² Since all functions are convex, just need to solve ± Quadratic ½ x T Qx + b T x ² Solve: Qx + b= 0 ² Convexity ÅÆ Q positive semi-definite ± All principal minors are non-negative = n i i i x f 1 ) ( 0 ) ( ' = i i x f
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MIE376 - Mathematical Programming 8 Steepest Ascent/Descent ± Assume smooth f(x 1 ,x 2 ,…,x n ) ± Start at arbitrary point x ± Determine direction of steepest ascent/descent ± Determine optimal step size using golden search ² Min f[ x + t f( x ) ], such that t>0 ± Take the step x Æ x + t* f( x ) ± If || f( x ) || ~ 0 stop, otherwise return to 2 nd step ( ) n x f x f x f f = / ... / / ) ( 2 1 x
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MIE376 - Mathematical Programming 9 Objective Function is NOT smooth or convex ± No guarantee of optimality ± Local optima can be pervasive ± Many techniques – few are attractive ± Nearly impossible to predict which method works best.
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MIE376 Lec 5 - NLPQPv2 - Non-linear Optimization...

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