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수치최적화

연세대학교
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페이스북 트위터 카카오톡 네이버밴드 링크복사

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공학 >컴퓨터ㆍ통신 >정보통신공학
강의학기

2012년 2학기

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26,232

For graduate student in science and engineering. This course covers numerical optimization. We will concentrate on convex

optimization. For such purpose we will briefly cover the convex theory.For unconstrained optimization, we study algorithm.After that we will consider applications, approximation and fitting, l1 minimization, for examples.

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CH 1: The introduction of this course and basic concepts

차시별 강의

1.	CH 1: The introduction of this course and basic concepts	The schedule for this semester. Reference book, requirments and evaluation method, some basic concepts
	CH 2: Convex sets	1.affine and convex sets; 2. some important examples; 3. Operations that preserve convexity; 4. Separating and supporting hyperplanes
2.	CH 3: Convex functions	1. basic properties and examples; 2.operations that preserve convexity; 3.
	CH 3: Convex functions	1.operations that preserve convexity; 2. Jensens inequality; 3. conjugate functions
3.	Ch4: Convex optimization problems	1. Optimization problems in standard form; 2. Convex optimization problem;
	Ch4: Convex optimization problems	1. Convex optimization problem; 2. equivalent convex problems
	Ch4: Convex optimization problems; CH5: Duality	1. Linear program; 2. QCQP; 3.second order cone programming; 4: Lagrange dual function; 5: Standard form Lp
4.	CH5: Duality	1: Lagrange dual problem; 2: weak and strong duality
	CH5: Duality	1: Geometric interpretation;
	CH5: Duality	1. Slaters constraint equation; 2. KKT conditions
5.	CH5: Duality	1. Saddle point interpretation; 2. Perturbation and sensitivity analysis
	CH5: Duality; CH6: Unconstrained minimization	1. Problem reformulations; 2. Strong convexity and implications
6.	CH6: Unconstrained minimization	1. Descent methods; 2. linear search types; 3. Gradient descent method
	CH6: Unconstrained minimization	1. Quadratic problem example; 2. Nonquadratic problem example; 3. Steeppest descent method
	CH6: Unconstrained minimization	1. Different norm in normalized steepest descent method; 2. Choice of norm;
7.	CH6: Unconstrained minimization	1. Newton step; 2. Newton decrement
	CH6: Unconstrained minimization	1. Newton method; 2. Classical convergence analysis
	CH6: Unconstrained minimization	1. Classical convergence analysis; 2. Damped Newton phase; 3. Implementation
8.	CH7: Equality constrained minimization	1. Equality constrained minimization; 2. Quadratic minimization
	CH7: Equality constrained minimization	1. Quadratic minimization; 2. Eliminating equality constraints; 3. Example
9.	CH7: Equality constrained minimization	1. Newton step; 2. Newton decrement; 3. Newton method with equality constraints; 4. Newton method and elimination; 5. Newton step at infeasible points
	CH7: Equality constrained minimization	1. Infeasible start Newton method; 2. Solving KKT system; 3.Analytic centering
10.	CH8: Iterior Point Method	1. Inequality constrained minimization; 2. Logrithmic barrier; 3. Central path
	CH8: Iterior Point Method	1. Dual points on central path; 2. Interpretation via KKT conditions; 3. Force field interpretation; 4. Barrier method; 5. Convergence analysis
11.	CH8: Iterior Point Method	1. Barrier method; 2. Convergence analysis; 3. Feasibility and phase I methods
	CH8: Iterior Point Method	1. Primal-dual interior point methods; 2. Interpretation of Newton step
	CH8: Iterior Point Method	1. L1 norm approximation
12.	CH9: Approximation and fitting	1. Norm approximation; 2. Penalty function approximation
	CH9: Approximation and fitting	1. Example; 2. Huber penaltry function; 3. Least norm problems
13.	CH9: Approximation and fitting	1. Signal reconstruction; 2. Quadratic smoothing example; 3. Total variation reconstruction example
14.	CH9: Statistical estimation	1. Maximum likelihood estimation; 2. Linear measurments wkth IID noise; 3. Exampes
	CH9: Statistical estimation	1. Homework 3 comments; 2. Final project explanation
15.	CH9: Statistic estimation; Geometric problems; Penalty barrier and augmented Lagrangian methods	1. Logistic regression; 2. Linear discrimination; 3. Roboust linear discrimination; 4. Quadratic penalty method
	CH9: Penalty barrier and augmented Lagrangian methods	1. Augmented Lagrangian method; 2. L1 penlaty function