凸優化課程-1
introduction
- mathematical optimization
- list squares and linear programming
- convex optimization
- example
- course goals and topics
- nonlinear optimization
- brief history of conves optimization
optimization problem
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objective function and optimization functoion
portfolio optimization 投資組合優化
- variables: amounts invested in different assets
- constraints: budget, investment per asset, minimum return
- objective : overall risk or return variance
device sizing in electronic circuits 電子線路的尺寸
- variables: device widths and lengths
- constraints: manufacting limits, timing requirements, maximum area
- objective : power consumption
data fitting 資料拟合 statistics estimation
- cariables: model parameters(不同之處)
- constraints: prior information, parameter limits
- objective: measure of misfit or prediction
some people use general optimization all the time
solving optimization problems (如何解決這類問題)
general optimization problem (一般的問題)
- very difficult to solve 十分難以解決
- methods involve some comprimise(very long time)
exceptions: certain problem can be solved efficiently and reliably
- least-squares problem 最小二乘問題
- linear programming problems 線性規劃問題
- convex optimization problems 凸優化問題
affine function 仿射函數
solving convex optimization problems
- no analytical solution
- reliable and efficient algorithms
- computation time (roughly) proportional to max{n3, n2m, F}, whereFis cost of evaluatingfi’s and their first and second derivatives
- almost a technology
using convex optimization
- often difficult to recognize
- many ticks for transforming problems into convex form
- surpridingly many problem can be solved via convex optimization
additional constraints: does adding 1 or 2 below complicate the problem?
- no more than half of the total power is in any 10 lamps
- no more than half of the lamps are on $ (p_j>0)$
goals
- recognize formulate problem as convex optimization problems
- develop code for problems of moderate size.
- charactierize optimal solution, give limits of performance, etc.
topics
- convex sets, functions, optimization problem
- example and application
- algorithms
traditional techniques for general nonconvex problems involve compremises