모두를 위한 컨백스 최적화

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convex-optimization-for-all.github.io's Introduction

모두를 위한 컨벡스 최적화

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OpenSource에 기여하고 싶다면

저자 서문

기계학습에 세간의 이목이 집중되며 최적화에 대한 관심도도 나날이 상승하고 있습니다. 허나, 입문자를 위한 한글자료가 풍부하지 않아 많은 분들이 그 진입장벽으로 힘들어하는 것에 안타까움을 느꼈습니다. 이에 모두의 연구소의 풀잎스쿨에 Convex Optimization 과정을 개설하였고, 지식 나눔을 실천하고자 하는 참여자분들의 선의의 의지에 힘입어 본 프로젝트를 시작하게 되었습니다. 이 활동을 통해 부디 전국민의 지적 성장과 컨벡스 최적화의 국내 대중화에 힘을 보탤수 있길 기원합니다.

이 문서의 전반적인 내용은 카네기멜론 대학 강의자료를 참고하였고, 보조 교재로는 스탠포드 대학 강의자료를 사용하였습니다. 본 ebook을 중심으로 두 강의자료를 레퍼런스로 공부하시면 좋습니다.

[email protected] / 박진우 (컨벡스 최적화 풀잎스쿨, 모두의 연구소)

옮긴이 서문

최근 머신러닝의 지속적인 발전 속에서 다양한 연구들이 진행되고 있고, 이를 현실 문제에 적용하려는 움직임 또한 커지고 있습니다. 하지만 머신러닝의 근간을 이루는 수학에 대한 심도 높은 이해가 없다면 그에 대한 이해와 적용 또한 피상적으로 이뤄질 수 밖에 없습니다.

Convex Optimization은 머신러닝과 직접적으로 연관이 많을 뿐더러 선형대수, 미적분학, 수치해석과 같이 수학의 다양한 하위 분야들을 포함하고 있다는 점에서 머신러닝을 공부하는 사람들에게 매력적인 학문입니다. 다만 홀로 다루기에는 내용이 적지 않을 뿐더러 학문 자체의 난이도도 높은 편이기에 함께 공부할 사람들을 모아 2021년 Convex Optimization Study를 시작하게 되었습니다. 본 Blog는 함께 진행한 Study의 흔적이자, 후에 혼자 공부하고자 하시는 분들께 도움을 드리고자 만들었습니다.

본 Blog의 주요 컨텐츠는 모두를 위한 컨벡스 최적화의 저자 분들의 동의를 구해 Migration한 내용들입니다. 원 컨텐츠는 Convex Optimization에 관한 한국어 컨텐츠 중 가장 잘 알려져 있으면서 내용적으로도 부족함이 없습니다. 본 Blog에서는 기존 WikiDocs 컨텐츠를 이어 받아 Open Source로 만들어 보고자 합니다. 따라서 누구나 컨텐츠에 이슈를 제기하고 직접 Pull Request를 생성하여 기여할 수 있습니다. 이를 통해 모두를 위한 컨벡스 최적화 저자분들의 뜻이기도 한 '전국민의 지적 성장과 컨벡스 최적화의 국내 대중화'에 작은 보탬이 될 수 있기를 바랍니다.

[email protected] / 우경민 (마키나락스)

다루는 내용들

idx	Title	Book	Lecture	Slide
1	Introduction	Page	CMU Lecture	CMU Note
2	Convex Sets	Page	Stanford Lecture	Stanford Note
3	Convex Functions	Page	Stanford Lecture	Stanford Note
4	Convex Optimization Basis	Page	CMU Lecture	CMU Note
5	Canonical Problems	Page	CMU Lecture	CMU Note
6	Gradient Descent	Page	CMU Lecture	CMU Note
7	Subgradient	Page	CMU Lecture	CMU Note
8	Subgradient Method	Page	CMU Lecture	CMU Note
9	Proximal Gradient Descent and Acceleration	Page	CMU Lecture	CMU Note
10	Duality in Linear Programs	Page	CMU Lecture	CMU Note
11	Duality in General Programs	Page	CMU Lecture	CMU Note
12	KKT Conditions	Page	CMU Lecture	CMU Note
13	Duality uses and correspondences	Page	CMU Lecture	CMU Note
14	Newton's Method	Page	CMU Lecture	CMU Note
15	Barrier Method	Page	CMU Lecture	CMU Note
16	Duality Revisited	Page	CMU Lecture	CMU Note
17	Primal-Dual Interior-Point Methods	Page	CMU Lecture	CMU Note
18	Quasi-Newton Methods	Page	CMU Lecture	CMU Note
19	Proximal Netwon Method	Page	CMU Lecture	CMU Note
20	Dual Methods	Page	CMU Lecture	CMU Note
21	Alternating Direction Method of Mulipliers	Page	CMU Lecture	CMU Note
22	Conditional Gradient Method	Page	CMU Lecture	CMU Note
23	Coordinate Descent	Page	CMU Lecture	CMU Note
24	Mixed Integer Programming 1	Page	CMU Lecture	CMU Note
25	Mixed Integer Programming 2	Page	CMU Lecture	CMU Note

참고한 자료들

원저자 및 리뷰어

원저자 (사전순)

김기범([email protected])
김정훈([email protected])
노원종([email protected])
박진우([email protected])
윤성진([email protected])
이규복([email protected])
한영일([email protected])
황혜진([email protected])

리뷰어 (사전순)

이주희 ([email protected])
장승환 ([email protected])
정태수 ([email protected])

원저자 및 리뷰어 상세소개

테마

Lanyon by Mark Otto

Contributors ✨

Thanks goes to these wonderful people (emoji key):

_{KyeongMin WOO} 💻	_{Wontak Ryu} 💻	_LEEMINJOO 💻	_{HoonCheol Shin} 💻	_{Jinwoo Park (Curt)} 💻	_{YoungJaeChoung} 💻	_{Kibum Fred Kim} 💻
_{Eugene Yang} 🐛	_{Seongjin Kim} 💻	_{Ham Ji Seong} 💻	_seolhokim 📖 🚧

This project follows the all-contributors specification. Contributions of any kind welcome!

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chapter01/_posts/21-01-28-convex_optimization_problem.md:66:![](https://wikidocs.net/images/page/17206/epigraph.png)
chapter01/_posts/21-01-28-goals_and_topics.md:30:![](https://wikidocs.net/images/page/17206/2d_fused_lasso.png)
chapter01/_posts/21-01-28-goals_and_topics.md:45:![](https://wikidocs.net/images/page/17208/result1.png)
chapter01/_posts/21-01-28-goals_and_topics.md:48:![](https://wikidocs.net/images/page/17208/result2.png)
chapter01/_posts/21-01-28-goals_and_topics.md:51:![](https://wikidocs.net/images/page/17208/result3.png)
chapter02/_posts/21-02-11-02_06_02_Dual_generalized_inequalities.md:49:  <img src="https://wikidocs.net/images/page/17375/02.06_02_Minimum_element.PNG" alt="[Fig1] Minimum element [1]" width="70%">
chapter02/_posts/21-02-11-02_06_02_Dual_generalized_inequalities.md:65:  <img src="https://wikidocs.net/images/page/17375/02.06_05_Minimal_element.PNG" alt="[Fig2] Minimal element [1]" width="70%">
chapter02/_posts/21-02-11-02_06_02_Dual_generalized_inequalities.md:74:  <img src="https://wikidocs.net/images/page/17423/02.06_06_Minimal_element2.PNG" alt="[Fig3] Minimal이지만 minimizer가 아닌 예 [1]" width="70%">
chapter02/_posts/21-02-11-02_06_02_Dual_generalized_inequalities.md:83:  <img src="https://wikidocs.net/images/page/17423/02.06_07_Minimal_element3.PNG" alt="[Fig4] $$\lambda_1 \succ_K^* 0$$으로 강화되지 않는 Minimal 예  [1]" width="70%">
chapter02/_posts/21-02-11-02_06_02_Dual_generalized_inequalities.md:107:  <img src="https://wikidocs.net/images/page/17423/02.06_08_Pareto_optimal.PNG" alt="[Fig5] Optimal production frontier [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:21:  <img src="https://wikidocs.net/images/page/17371/02.05_Hyperplane.PNG" alt="[Fig1] Hyperplane [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:36:  <img src="https://wikidocs.net/images/page/17371/02.06_Halfspace.PNG" alt="[Fig2] Halfspace [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:67:  <img src="https://wikidocs.net/images/page/17371/02.07_Ellipsoid.PNG" alt="[Fig3] Ellipsoid [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:93:  <img src="https://wikidocs.net/images/page/17371/02.07_2_norm_ball.png" alt="[Fig4] Norm ball [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:102:  <img src="https://wikidocs.net/images/page/17371/02.07_3_norm_ball2.PNG" alt="[Fig4] Norm ball [2]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:120:  <img src="https://wikidocs.net/images/page/17371/02.09_Polyhedra.PNG" alt="[Fig5] Polyhedra [1]" width="70%">
chapter02/_posts/21-02-11-02_02_01_Convex_sets_examples.md:158:  <img src="https://wikidocs.net/images/page/17371/02.02_10_Simplex.PNG" alt="[Fig6] Simplex [source - wikipedia]" width="70%">
chapter02/_posts/21-02-11-02_02_03_Operations_that_preserve_convexity.md:53:  <img src="https://wikidocs.net/images/page/17372/02.03_03_pine_hole.png" alt="[Fig 1] pin-hole 카메라 작동 원리" width="70%">
chapter02/_posts/21-02-11-02_02_03_Operations_that_preserve_convexity.md:63:  <img src="https://wikidocs.net/images/page/17372/02.03_04_pine_hole_camera_model.png" alt="[Fig 2] pin-hole 카메라의 perspective function [1]" width="70%">
chapter02/_posts/21-02-11-02_02_03_Operations_that_preserve_convexity.md:82:  <img src="https://wikidocs.net/images/page/17372/02.03_05_linear_fractional_function.PNG" alt="[Fig 3] Linear-fractional functions [1]" width="70%">
chapter02/_posts/21-02-08-02_01_04_Convex_cone.md:29:  <img src="https://wikidocs.net/images/page/17390/02.04_Convex_Cone.PNG" alt="[Fig1] Convex Cone [1]" width="70%">
chapter02/_posts/21-02-08-02_01_04_Convex_cone.md:56:  <img src="https://wikidocs.net/images/page/17390/02.04_1_Conic_hull.PNG" alt="[Fig2] Conic hull [1]" width="70%">
chapter02/_posts/21-02-08-02_01_03_Convex-set.md:24:  <img src="https://wikidocs.net/images/page/17384/02.02_Convex_Set.PNG" alt="[Fig1] Convex Set [1]" width="70%">
chapter02/_posts/21-02-08-02_01_03_Convex-set.md:50:  <img src="https://wikidocs.net/images/page/17384/02.03_Convex_Hull.PNG" alt="[Fig2] Convex hull [1]" width="70%">
chapter02/_posts/21-02-11-02_06_01_Dual_cones.md:20:  <img src="https://wikidocs.net/images/page/17422/02.06_01_2_dual_cone.PNG" alt="[Fig1] Dual cone 정의 구간" width="70%">
chapter02/_posts/21-02-11-02_06_01_Dual_cones.md:30:  <img src="https://wikidocs.net/images/page/17375/02.06_01_dual_cone.PNG" alt="[Fig2] Dual cone과 supporting hyperplanne의 normal[1]" width="70%">
chapter02/_posts/21-02-11-02_06_01_Dual_cones.md:50:  <img src="https://wikidocs.net/images/page/17375/02.06_03_L2_self-dual.PNG" alt="[Fig3] $$l_2$$ cone과 dual cone" width="70%">
chapter02/_posts/21-02-11-02_06_01_Dual_cones.md:62:  <img src="https://wikidocs.net/images/page/17375/02.06_04_L_inf_dual_norm.PNG" alt="[Fig4] $$l_\infty$$ cone과 dual cone" width="70%">
chapter02/_posts/21-02-11-02_02_02_Convex_cone_examples.md:21:  <img src="https://wikidocs.net/images/page/17371/02.08_Norm_Cone.PNG" alt="[Fig1] Norm Cone [1]" width="70%">
chapter02/_posts/21-02-11-02_02_02_Convex_cone_examples.md:44:  <img src="https://wikidocs.net/images/page/17371/02.04_2_Normal_Cone.PNG" alt="[Fig2] Normal Cone [3]" width="70%">
chapter02/_posts/21-02-11-02_02_02_Convex_cone_examples.md:69:  <img src="https://wikidocs.net/images/page/17371/02.10_Positive_Semidefinite_Cone.PNG" alt="[Fig3] Positive semidefinite cone [1]" width="70%">
chapter02/_posts/21-02-11-02_05_Separating_and_supporting_hyperplanes.md:21:  <img src="https://wikidocs.net/images/page/17374/02.05_01_Seperating_hyperplan_theorem.PNG" alt="[Fig1] Separating hyperplane theorem [1]" width="70%">
chapter02/_posts/21-02-11-02_05_Separating_and_supporting_hyperplanes.md:46:  <img src="https://wikidocs.net/images/page/17374/02.05_02_Supporting_hyperplane_theorem.PNG" alt="[Fig 2] Supporting hyperplane [1]" width="70%">
chapter02/_posts/21-02-11-02_04_Generalized_inequalities.md:78:  <img src="https://wikidocs.net/images/page/17373/02.06_01_Minimum_and_minimal.PNG" alt="[Fig1] Minimum and minimal elements [1]" width="70%">
chapter02/_posts/21-02-08-02_01_01_Line_line segment_ray.md:15:  <img src="https://wikidocs.net/images/page/17382/02.01_Line_Segment.PNG" alt="Line Segment" width="70%">
chapter03/_posts/21-02-12-03_01_01_convex_functions_definition.md:18: <img src="https://wikidocs.net/images/page/17495/convex_function01.png" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_03_the_conjugate_function.md:16: <img src="https://wikidocs.net/images/page/17428/conjugate_function.png" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_04_01_quasiconvex_functions_definition_and_examples.md:24: <img src="https://wikidocs.net/images/page/17416/Fig3.9_quasiconvex_ftn_cAsoUpr.PNG" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_01_03_key_properties_of_convex_functions.md:34: <img src="https://wikidocs.net/images/page/17269/1st_order_condition.png" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_01_03_key_properties_of_convex_functions.md:66: <img src="https://wikidocs.net/images/page/17497/jensen_inequality.png" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_04_03_differentiable_quasiconvex_functions.md:16: <img src="https://wikidocs.net/images/page/17418/3.12_Three_level_curves_OV6vtPq.PNG" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_04_02_basic_properties_of_quasiconvex_functions.md:19: <img src="https://wikidocs.net/images/page/17419/Fig.3.10_quasiconvex_function_on_R_4uChnEm.PNG" alt="" width="70%" height="70%">
chapter03/_posts/21-02-12-03_04_02_basic_properties_of_quasiconvex_functions.md:35: <img src="https://wikidocs.net/images/page/17419/Fig.3.11_quasiconvex_function_on_R_2_PPQpNiU.PNG" alt="" width="70%" height="70%">
chapter04/_posts/20-02-08-04_02_Convex_solution_sets.md:48:  <img src="https://wikidocs.net/images/page/18263/multiple-optima.png" alt="[Fig1] geometric interpretation of convexity of the solution set">
chapter04/_posts/20-02-08-04_03_First_order_optimality_condition.md:28:  <img src="https://wikidocs.net/images/page/18337/first-order-condition.png" alt="[Fig1] geometric interpretation of first-order condition for optimality [3]">
chapter04/_posts/20-02-08-04_04_Partial_optimization.md:15:  <img src="https://wikidocs.net/images/page/18367/partial-optimization.png" alt="[Fig1] partial optimization of a convex problem [3]">
chapter05/_posts/21-02-08-05_04_Second_Order_Cone_Programming_(SOCP).md:31:  <img src="https://wikidocs.net/images/page/17371/02.08_Norm_Cone.PNG" alt="[Fig1] Norm Cone [1]" width="70%">
chapter05/_posts/21-02-08-05_01_Linear_Programming_(LP).md:27:  <img src="https://wikidocs.net/images/page/17850/geometric_interpretation_of_LP.png" alt="[Fig1] Geometric interpretation of LP [1]" width="70%">
chapter05/_posts/21-02-08-05_00_Canonical_Problems.md:15:  <img src="https://wikidocs.net/images/page/17203/Optimization_problem.png" alt="[Fig1] Convex Optimization Problem in standard form [3]" width="70%">
chapter05/_posts/21-02-08-05_00_Canonical_Problems.md:38:  <img src="https://wikidocs.net/images/page/17851/canonical_problems.jpg" alt="[Fig2] Canonical Problems" width="90%">
chapter05/_posts/21-02-08-05_02_Quadratic_Programming_(QP).md:28:  <img src="https://wikidocs.net/images/page/17852/geometric_interpretation_of_QP.png" alt="[Fig 1] Geometric interpretation of QP [1]" width="70%">
chapter06/_posts/21-03-20-06_04_gradient_boosting.md:36:  <img src="https://wikidocs.net/images/page/19037/tree_O9zyAlk.png" alt="tree_O9zyAlk" width="80%" height="80%">
chapter06/_posts/21-03-20-06_03_04_convergence_analysis_under_strong_convexity.md:122:  <img src="https://wikidocs.net/images/page/18092/06.03_01_01_Line_Convergence.PNG" alt="Line_Convergence" width="60%" height="60%">
chapter06/_posts/21-03-20-06_01_gradient_descent.md:37:  <img src="https://wikidocs.net/images/page/18084/gradientdescent1.PNG" alt="gradientdescent1" width="80%" height="80%">
chapter06/_posts/21-03-20-06_01_gradient_descent.md:46:  <img src="https://wikidocs.net/images/page/18084/gradientdescent2.PNG" alt="gradientdescent2" width="80%" height="80%">
chapter06/_posts/21-03-20-06_01_gradient_descent.md:74:  <img src="https://wikidocs.net/images/page/18084/gradientdescent3.PNG" alt="gradientdescent3" width="80%" height="80%">
chapter06/_posts/21-03-20-06_02_02_backtracking_line_search.md:18:  <img src="https://wikidocs.net/images/page/18184/06.02_02_01_Backtracking_Line_Search.PNG" alt="backtrackinglinesearch1" width="100%" height="100%">
chapter06/_posts/21-03-20-06_02_02_backtracking_line_search.md:48:  <img src="https://wikidocs.net/images/page/18184/06.02_02_02_Convergence.PNG" alt="backtrackinglinesearch1" width="70%" height="70%">
chapter06/_posts/21-03-20-06_02_02_backtracking_line_search.md:74:  <img src="https://wikidocs.net/images/page/18184/f_leq_app.png" alt="f_leq_app" width="60%" height="60%">
chapter06/_posts/21-03-20-06_02_02_backtracking_line_search.md:85:  <img src="https://wikidocs.net/images/page/18184/f_geq_app.png" alt="f_geq_app" width="60%" height="60%">
chapter06/_posts/21-03-20-06_02_01_fixed_step_size.md:17:  <img src="https://wikidocs.net/images/page/18088/gradientdescent4.PNG" alt="gradientdescent4" width="100%" height="100%">
chapter07/_posts/21-03-25-07_03_04_example_soft-thresholding.md:43: <img src="https://wikidocs.net/images/page/18963/subgrad-6.png" alt="connection_to_convexity_geometry" width="80%" height="80%">
chapter07/_posts/21-03-25-07_01_subgradient.md:35:  <img src="https://wikidocs.net/images/page/18963/subgrad-1.png" alt="Subgradient1" width="80%" height="80%">
chapter07/_posts/21-03-25-07_01_subgradient.md:55:  <img src="https://wikidocs.net/images/page/18963/subgrad-3.png" alt="Subgradient2" width="80%" height="80%">
chapter07/_posts/21-03-25-07_01_subgradient.md:75:  <img src="https://wikidocs.net/images/page/18963/subgrad-2.png" alt="Subgradient3" width="80%" height="80%">
chapter07/_posts/21-03-25-07_01_subgradient.md:91:  <img src="https://wikidocs.net/images/page/18963/subgrad-4.png" alt="Subgradient4" width="80%" height="80%">
chapter07/_posts/21-03-25-07_02_01_connection_to_a_convexity_geometry.md:71:  <img src="https://wikidocs.net/images/page/18963/subgrad-5.png" alt="connection_to_convexity_geometry" width="80%" height="80%">
chapter08/_posts/20-03-29-08_02_04_batch_vs_stochastic_methods.md:19:  <img src="https://wikidocs.net/images/page/18973/stochastic_vs_batch.PNG" alt="stochastic_vs_batch" width="80%" height="80%">
chapter08/_posts/20-03-29-08_01_05_example_regularized_logistic_regression.md:34:  <img src="https://wikidocs.net/images/page/19145/grad_vs_subgrad.PNG" alt="grad_vs_subgrad" width="90%" height="90%">
chapter08/_posts/20-03-29-08_01_07_example_intersection_of_sets.md:84:  <img src="https://wikidocs.net/images/page/18975/projection.PNG" alt="projection" width="60%" height="60%">
chapter09/_posts/20-01-08-09_05_01_accelerated_proximal_gradient_method.md:49:  <img src="https://wikidocs.net/images/page/19403/momentum_weight.png" width="80%" height="80%">
chapter09/_posts/20-01-08-09_05_01_accelerated_proximal_gradient_method.md:62:  <img src="https://wikidocs.net/images/page/19403/accelerated_proximal_gradient.png" width="80%" height="80%">
chapter09/_posts/20-01-08-09_01_proximal_gradient_descent.md:113:  <img src="https://wikidocs.net/images/page/19032/09.01_01_ISTA.PNG" width="80%" height="80%">
chapter09/_posts/20-01-08-09_05_03_example_FISTA.md:31:  <img src="https://wikidocs.net/images/page/19403/FISTA.png" width="80%" height="80%">
chapter09/_posts/20-01-08-09_05_03_example_FISTA.md:40:  <img src="https://wikidocs.net/images/page/19403/FISTA2.png" width="80%" height="80%">
chapter09/_posts/20-01-08-09_04_special_cases.md:50:  <img src="https://wikidocs.net/images/page/20230/projected_gradient_descent.png" width="80%" height="80%">
chapter10/_posts/21-03-22-10_03_Max_flow_and_min_cut.md:22: <img src="https://wikidocs.net/images/page/20588/max_flow.png" alt="" width="70%" height="70%">
chapter10/_posts/21-03-22-10_03_Max_flow_and_min_cut.md:77: <img src="https://wikidocs.net/images/page/20588/min_cut.png" alt="" width="70%" height="70%">
chapter10/_posts/21-03-22-10_05_Matrix_Games.md:21:  <img src="https://wikidocs.net/images/page/19936/matrix_game.png" alt="Line Segment" width="70%">
chapter11/_posts/21-03-24-11_3_Lagrange_dual_problem.md:47:  <img src="https://wikidocs.net/images/page/20584/dual-gen_13.png" width="70%">
chapter11/_posts/21-03-24-11_2_Lagrange_dual_function.md:24:  <img src="https://wikidocs.net/images/page/20583/dual-gen_7.png" width="70%">
chapter11/_posts/21-03-24-11_2_Lagrange_dual_function.md:132:  <img src="https://wikidocs.net/images/page/20583/dual-gen_10.png" width="70%">
chapter11/_posts/21-03-24-11_1_Lagrangian.md:49:  <img src="https://wikidocs.net/images/page/20579/dual-gen_6.PNG" width="70%">
chapter12/_posts/21-04-02-12_04_Example_support_vector_machines.md:46: <img src="https://wikidocs.net/images/page/20962/svm.png" alt="" width="70%" height="70%">
chapter12/_posts/21-04-02-12_06_Uniqueness_in_L1_penalized_problems.md:143: <img src="https://wikidocs.net/images/page/20964/l1_uniqueness.png" alt="" width="70%" height="70%">
chapter12/_posts/21-04-02-12_05_Constrained_and_Lagrange_forms.md:64: <img src="https://wikidocs.net/images/page/20963/conclusion.png" alt="" width="70%" height="70%">
chapter12/_posts/21-04-02-12_03_Example_water_filling.md:60: <img src="https://wikidocs.net/images/page/20961/water-fill.png" alt="" width="70%" height="70%">
chapter13/_posts/21-04-05-13_05_Dual_cones.md:20: <img src="https://wikidocs.net/images/page/21005/dual_cone.png" alt="" width="70%" height="70%">
chapter13/_posts/21-04-05-13_04_01_Example_lasso_dual.md:74: <img src="https://wikidocs.net/images/page/21003/Conjugate_LassoDual_Example.png" alt="" width="70%" height="70%">
chapter13/_posts/21-04-05-13_04_Conjugate_function.md:16: <img src="https://wikidocs.net/images/page/21001/conjugate_function.png" alt="" width="70%" height="70%">
chapter14/_posts/2021-03-26-14_04_backtracking_line_search.md:43: <img src="https://wikidocs.net/images/page/21334/2.jpg" alt="" width="70%" height="70%">
chapter14/_posts/2021-03-26-14_07_comparison_to_first_order_method.md:28: <img src="https://wikidocs.net/images/page/21755/gd.JPG" alt="" width="70%" height="70%">
chapter14/_posts/2021-03-26-14_02_01_root_finding.md:68: <img src="https://wikidocs.net/images/page/21332/table1.JPG" alt="" width="90%">
chapter14/_posts/2021-03-26-14_02_03_local_convergence_analysis.md:55: <img src="https://wikidocs.net/images/page/21708/1_.png" alt="" width="70%" height="70%">
chapter14/_posts/2021-03-26-14_01_01_newton_method_interpretation.md:67: <img src="https://wikidocs.net/images/page/21331/gd.JPG" alt="" width="70%" height="70%">
chapter15/_posts/21-03-28-15_02_central_path.md:37: <img src="https://wikidocs.net/images/page/21298/15_central_path_02.PNG" alt="" width="70%" height="70%">
chapter15/_posts/21-03-28-15_06_barrier_method_v2.md:38: <img src="https://wikidocs.net/images/page/21320/15_barrier_methodv2_04.PNG" alt="" width="70%" height="70%">
chapter15/_posts/21-03-28-15_06_barrier_method_v2.md:48: <img src="https://wikidocs.net/images/page/21320/15_barrier_methodv2_05.PNG" alt="" width="70%" height="70%">
chapter15/_posts/21-03-28-15_01_02_log_barrier_function_and_barrier_method.md:20: <img src="https://wikidocs.net/images/page/21305/15_barrier_function_01.PNG" alt="" width="70%" height="70%">
chapter15/_posts/21-03-28-15_04_barrier_method_v0_and_v1.md:59: <img src="https://wikidocs.net/images/page/21300/15_barrier_method_03.PNG" alt="" width="70%" height="70%">
chapter17/_posts/21-05-01-17_04_special_case_linear_programming.md:134:  <img src="https://wikidocs.net/images/page/21647/barrier_vs_primal_dual.png">
chapter17/_posts/21-05-01-17_04_special_case_linear_programming.md:149:  <img src="https://wikidocs.net/images/page/21647/barrier_vs_primal_dual2.png">
chapter18/_posts/21-03-23-18_07_Limited_Memory_BFGS_(LBFGS).md:45:  <img src="https://wikidocs.net/images/page/22155/algorithm_quasi-newton.png" alt="[Fig1] The algorithm of LBFGS [3]" width="90%">
chapter18/_posts/21-03-23-18_01_Secant_Equation_and_Curvature_Condition.md:30:  <img src="https://wikidocs.net/images/page/22150/intuition_of_secant_eq.png" alt="[Fig1] The intuition of secant equation" width="70%">
chapter19/_posts/21-03-24-19_01_03_Scaled_proximal_map.md:30:  <img src="https://wikidocs.net/images/page/22434/09.01_03_projection_operator.PNG" alt="[Fig 1] Projection onto a convex set C [3]" width="70%">
chapter19/_posts/21-03-24-19_01_01_Reminder:_proximal_gradient_descent.md:62:  <img src="https://wikidocs.net/images/page/22431/09.01_01_proximal_gradient_descent.PNG" alt="[Fig 1] Proximal gradient descent updates [3]" width="70%">
chapter19/_posts/21-03-24-19_05_Notable_examples.md:31:  <img src="https://wikidocs.net/images/page/22428/09.05_Lasso_Example1.PNG" alt="[Fig 1] Dense hessian X (n=5000, p=6000) [2]" width="70%">
chapter19/_posts/21-03-24-19_05_Notable_examples.md:46:  <img src="https://wikidocs.net/images/page/22428/09.05_Lasso_Example-sparse.PNG" alt="[Fig 2] Sparse hessian X (n=542,000, p=47,000) [2]" width="70%">
chapter19/_posts/21-03-24-19_05_Notable_examples.md:61:  <img src="https://wikidocs.net/images/page/22428/09.05_Inexact_prox.PNG" alt="[Fig 3] Three stopping rules [2]" width="70%">
chapter20/_posts/21-03-27-20_02_01_Dual_Decomposition_with_Equality_Constraint.md:39:  <img src="https://wikidocs.net/images/page/23703/decomposition.png" alt="[Fig 1] Broadcast and Gather" width="70%">
chapter21/_posts/21-03-29-21_04_Example_:_Sparse_subspace_estimation_and_sparse_plus_low_rank_decomposition.md:18:이 문제는 projection 행렬의 set이 convex set이 아니기 때문에 non-convex 문제이다. 하지만, 아래의 convex 문제와 동일함이 알려져 있다.[[VCLR13](https://wikidocs.net/22687)] 이는 subspace estimation problem이라고도 불린다.
chapter21/_posts/21-03-29-21_04_Example_:_Sparse_subspace_estimation_and_sparse_plus_low_rank_decomposition.md:59:여기서 $$P_{F_{k}}$$는 fantope projection operator이다. 이는 eigendecomposition $$A= U\sum U^{T}, \sum = diag(\sigma_{1},...\sigma_{p})$$의  clipping으로 정의된다.[[VCLR13](https://wikidocs.net/22687)]:
chapter21/_posts/21-03-29-21_04_Example_:_Sparse_subspace_estimation_and_sparse_plus_low_rank_decomposition.md:69:$$M\in \mathbb{R}^{n\times m}$$일때, sparse plue low rank decomposition problem은 다음과 같다.[[CLMW09](https://wikidocs.net/edit/page/22687)]
chapter21/_posts/21-03-29-21_04_Example_:_Sparse_subspace_estimation_and_sparse_plus_low_rank_decomposition.md:90:  <img src="https://wikidocs.net/images/page/24031/candes.png" alt="[Fig 1] Example of sparse plue low rank decomoposition on surveliance camera[3]" width="70%">
chapter21/_posts/21-03-29-21_06_Faster_convergence_with_subprogram_parametrization_:_example_of_the_2d_fused_lasso_problem.md:27:  <img src="https://wikidocs.net/images/page/24033/2dfussed.png" alt="[Fig 1] Interpretation of the penalty term in 2d fussed lasso[3]" width="70%">
chapter21/_posts/21-03-29-21_06_Faster_convergence_with_subprogram_parametrization_:_example_of_the_2d_fused_lasso_problem.md:86:  <img src="https://wikidocs.net/images/page/24033/2dfussedlasso.png" alt="[Fig 2]  Interpretation of the matrix form ADMM updates for 2d fused lasso[3]" width="70%">
chapter21/_posts/21-03-29-21_06_Faster_convergence_with_subprogram_parametrization_:_example_of_the_2d_fused_lasso_problem.md:99:  <img src="https://wikidocs.net/images/page/24033/ll1.png" alt="[Fig 3]  Data, exact solution image(300x200 image : n = 60,000).
chapter21/_posts/21-03-29-21_06_Faster_convergence_with_subprogram_parametrization_:_example_of_the_2d_fused_lasso_problem.md:109:  <img src="https://wikidocs.net/images/page/24033/ll2.png" alt="[Fig 4]  Convergence curves of two ADMM algorithms. black : standard(vector form), red : specialized(matrix form) [3]" width="70%">
chapter21/_posts/21-03-29-21_06_Faster_convergence_with_subprogram_parametrization_:_example_of_the_2d_fused_lasso_problem.md:117:  <img src="https://wikidocs.net/images/page/24033/ll2.png" alt="[Fig 5]  ADMM iterates visualized after k = 10, 30, 50, 100 iterations [3]" width="70%">
chapter21/_posts/21-03-29-21_03_Example_:_Lasso_regression_and_group_lasso_Regression.md:19:이전의 여러 장에서, 우리는 lasso 문제를 여러가지 방법으로 해결해보았다. 대표적으로는 [proximal gradient descent(ISTA)](https://wikidocs.net/19032), [accelerated proximal gradient descent(FISTA)](https://wikidocs.net/20247), [barrier method](https://wikidocs.net/21297), [primal-dual interior-point method](https://wikidocs.net/21616) 등이 있다.
chapter21/_posts/21-03-29-21_03_Example_:_Lasso_regression_and_group_lasso_Regression.md:49:  <img src="https://wikidocs.net/images/page/24034/lasso.png" alt="[Fig 1] Comparison of various algorithms for lasso regression (50 instances with n = 100, p = 20) [3]" width="70%">
chapter22/_posts/21-03-28-22_02_Conditional_gradient_method.md:67:  <img src="https://wikidocs.net/images/page/22689/frank_wolfe.png" alt="[Fig 1] Conditional Gradient (Frank-Wolfe) method (From Jaggi 2011)[3]">
chapter22/_posts/21-03-28-22_02_Conditional_gradient_method.md:159:  <img src="https://wikidocs.net/images/page/22689/comparing_projected_and_conditional_gradient.png" alt="[Fig 2] Comparing projected and conditional gradient for constrained lasso
chapter22/_posts/21-03-28-22_02_Conditional_gradient_method.md:168:![](https://wikidocs.net/images/page/22689/comparing_projected_and_conditional_gradient.png)
chapter22/_posts/21-03-28-22_04_Properties_and_variants.md:23:  <img src="https://wikidocs.net/images/page/22690/away_steps.png" alt="[Fig 3] Away step motivation [3]">
chapter23/_posts/21-03-28-23_01_Coordinate_descent.md:23:  <img src="https://wikidocs.net/images/page/23401/smooth_function.png" alt="[Fig1] Smooth convex function f [3]">
chapter23/_posts/21-03-28-23_01_Coordinate_descent.md:39:  <img src="https://wikidocs.net/images/page/23401/non-smooth_function.png" alt="[Fig2] Counterexample: Non-smooth convex function f [3]">
chapter23/_posts/21-03-28-23_01_Coordinate_descent.md:74:  <img src="https://wikidocs.net/images/page/23401/separable_non-smooth.png" alt="[Fig3] Convex function f with separable non-smooth parts [3]">
chapter23/_posts/21-03-28-23_02_Example_linear_regression.md:25:  <img src="https://wikidocs.net/images/page/23401/smooth_function.png" alt="[Fig1] Smooth convex function f [3]">
chapter23/_posts/21-03-28-23_02_Example_linear_regression.md:39:  <img src="https://wikidocs.net/images/page/23401/non-smooth_function.png" alt="[Fig2] Counterexample: Non-smooth convex function f [3]">
chapter23/_posts/21-03-28-23_02_Example_linear_regression.md:74:  <img src="https://wikidocs.net/images/page/23401/separable_non-smooth.png" alt="[Fig3] Convex function f with separable non-smooth parts [3]">
chapter23/_posts/21-03-28-23_03_Example_lasso_regression.md:34:  <img src="https://wikidocs.net/images/page/23403/pd_vs_agd_vs_cd.png" alt="[Fig1] PD vs AGD vs CD [3]">
chapter25/_posts/21-03-28-25_02_02_Least_mean_squares.md:79:  <img src="https://wikidocs.net/images/page/23721/09.01_06_LQS_results1.PNG" alt="[Fig1] Mixed integer programming gap [3]">
chapter25/_posts/21-03-28-25_02_02_Least_mean_squares.md:88:  <img src="https://wikidocs.net/images/page/23721/09.01_07_LQS_results2.PNG" alt="[Fig2] Cold vs Warm Starts [3]">
chapter25/_posts/21-03-28-25_01_02_Cutting_plane_algorithm.md:53:![](https://wikidocs.net/images/page/23740/09.01_02_valid_inequality.PNG) <br>
chapter25/_posts/21-03-28-25_01_01_Convexification.md:35:  <img src="https://wikidocs.net/images/page/23719/09.01_01_cutting_plane_concept.PNG" alt="[Fig1] Cutting Plane">
chapter25/_posts/21-03-28-25_02_01_Best_subset_selection.md:80:  <img src="https://wikidocs.net/images/page/23722/09.01_03_subset_results1.PNG" alt="[Fig1] Dataset n=350, p = 64, k=6 [3]">
chapter25/_posts/21-03-28-25_02_01_Best_subset_selection.md:91:  <img src="https://wikidocs.net/images/page/23722/09.01_04_subset_results2.PNG" alt="[Fig2] Cold and Warm Starts [3]">
chapter25/_posts/21-03-28-25_02_01_Best_subset_selection.md:103:  <img src="https://wikidocs.net/images/page/23722/09.01_05_subset_results3.PNG" alt="[Fig3] Sparsity Detection (synthetic database) [3]">
chapter25/_posts/21-03-28-25_01_Cutting_planes.md:16:  <img src="https://wikidocs.net/images/page/23719/09.01_01_cutting_plane_concept.PNG" alt="[Fig1] Cutting Plane">

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24-01 Definition · 모두를 위한 컨벡스 최적화

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03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

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03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

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[migration] jekyll 설치작업이 혼란스러울수 있음.

jekyll 설치하는 과정과 버전에 대한 이슈가 발생할 수 있어 보입니다.

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03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

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[Closed] 03-04-04 Operations that preserve quasiconvexity

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[Closed] [Closed] 01-01 Optimization problems?

01-01 Optimization problems? · 모두를 위한 컨벡스 최적화

https://convex-optimization-for-all.github.io/chapter01/2021/01/07/optimization_problems/

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[Closed] 03-06 Convexity with respect to generalized inequalities

03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

http://127.0.0.1:4000/chapter03/2021/02/12/03_06_convexity_with_respect_to_generalized_inequalities/

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PR #58

01-04 Brief history of convex optimization · 모두를 위한 컨벡스 최적화

http://localhost:4000/chapter01/2021/01/28/brief_history_of_convex_optimization/

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[Closed] Chapter 08 과 09 포스트 날짜 2020년으로 표시됨

[Closed] 09-02 Convergence analysis

09-02 Convergence analysis · 모두를 위한 컨벡스 최적화

https://convex-optimization-for-all.github.io/chapter09/2020/01/08/09_02_convergence_analysis/

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01-02 Convex optimization problem · 모두를 위한 컨벡스 최적화

Command

Result

24-01 Definition · 모두를 위한 컨벡스 최적화

03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

03-04-04 Operations that preserve quasiconvexity · 모두를 위한 컨벡스 최적화

01-01 Optimization problems? · 모두를 위한 컨벡스 최적화

10-03 Max flow and min cut · 모두를 위한 컨벡스 최적화

01-01 Optimization problems? · 모두를 위한 컨벡스 최적화

01-01 Optimization problems? · 모두를 위한 컨벡스 최적화

03-01-01 Definition · 모두를 위한 컨벡스 최적화

01-01 Optimization problems? · 모두를 위한 컨벡스 최적화

03-06 Convexity with respect to generalized inequalities · 모두를 위한 컨벡스 최적화

01-04 Brief history of convex optimization · 모두를 위한 컨벡스 최적화

09-02 Convergence analysis · 모두를 위한 컨벡스 최적화

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