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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
MIE376 - Mathematical Programming 3 Introduction OR problems usually deal with man-made systems often ignore uncertainty involve linear and integer domains Æ LP and IP are all we need Sometimes Involve elements from the natural world Other sources of non-linearity 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.
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
MIE376 - Mathematical Programming 7 Some types of objectives… Separable Since all functions are convex, just need to solve Quadratic ½ x T Qx + b T x Solve: Qx + b= 0 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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