[工學(xué)]系統(tǒng)優(yōu)化調(diào)度-在線(xiàn)瀏覽
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【正文】 Late assignments will be discounted 10% a day, up to 5 days. This policy will be strictly enforced. – Homework solutions will be provided on the web sites a week after the due date. – Comments and discussions are encouraged in class, after class, during office hours or by appointment. – NO CHEATING! 9/11/2021 Fall 2021, Copy Right P. B. Luh 9 ? Reading Assignment: Bertsekas Sections , , and Appendices A and B ? Today: – Introduction and Unconstrained Optimization ? Motivation and Course Overview ? Problem Classification ? Optimality Conditions for Unconstrained Optimization ? Gradient Methods ? Framework ? Tomorrow: Bertsekas Sections ~ 9/11/2021 Fall 2021, Copy Right P. B. Luh 10 Mathematical Optimization Concepts and Algorithms Problem Classification ? General Formulation Minimize f(x), subject to x ? X ? Rn f(x): Cost function x: Decision variable X: Constraint set ? Example 1 – A small shop specializes in making 2 types of auto parts 9/11/2021 Fall 2021, Copy Right P. B. Luh 11 P1 P2 H rs . A v a i l a b l e Ca s t i n g 1 h r. 5 h rs . 1 6 0 h rs . D ri l l i n g 1 h r. 1 . 5 h rs . 6 0 h rs . F i n i s h i n g 2 h rs . 1 h r. 8 0 h rs . P ro fi t $ 3 0 $ 4 0 – How to decide the best quantities? What is the model? Max x1, x2 30 x1 + 40 x2, or Min x1, x2 ?(30 x1 + 40 x2), subject to x1 + 5x2 ? 160, (C1) x1 + ? 60, (C2) 2x1 + x2 ? 80, and (C3), x1 ? 0, x2 ? 0 (now treated as continuous variable) – What kind of problems is this? 9/11/2021 Fall 2021, Copy Right P. B. Luh 12 ? Linear cost function with linear constraints ? A Linear Programming (LP) problem ? Generic LP formulation Minimize f(x) ? c39。x subject to x ? X ? {x ? Rn, Ax = b, x ? 0} with c ? Rn, b ? Rm, A ? Rmxn given 9/11/2021 Fall 2021, Copy Right P. B. Luh 13 ? Example 2 – Same as Example 1 except that P1 P2 O r i g i n a l p r o f i t 3 0x 1 4 0x 2 N ew p r o f i t ( 5 0 – x 1 ) x 1 ( 6 4 – x 2 ) x 2 P rof itf or P 1x 1Max x1, x2 (50 – x1) x1 + (64 – x2) x2 subject to the same set of constraints ? Nonlinear cost function and/or nonlinear constraints ? Nonlinear programming (NLP) ~ with diminishing return 9/11/2021 Fall 2021, Copy Right P. B. Luh 14 ? Other types of problems: – Integer Programming (IP) ? With integer variables, ., if x1 and x2 are integers, or manufacturing scheduling problems to be discussed later – Mixed Integer Programming (MIP) – With both integer and continuous variables, ., power system unit mitment and economic dispatch ? Minimize the total generation cost ? Subject to system demand and reserve constraints ? By selecting unit up/down and generation levels ? Although many lectures will be on nonlinear programming, many results will be applicable to integer programming and mixed integer programming 9/11/2021 Fall 2021, Copy Right P. B. Luh 15 Optimality Conditions for Unconstrained Optimization ? Problem Formulation
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