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6.253 Convex Analysis and Optimization (MIT) 6.253 Convex Analysis and Optimization (MIT)

Description

6.253 develops the core analytical issues of continuous optimization, duality, and saddle point theory, using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions is discussed in detail, and is the basis for an intuitive, highly visual, geometrical approach to the subject. 6.253 develops the core analytical issues of continuous optimization, duality, and saddle point theory, using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions is discussed in detail, and is the basis for an intuitive, highly visual, geometrical approach to the subject.

Subjects

affine hulls | affine hulls | recession cones | recession cones | global minima | global minima | local minima | local minima | optimal solutions | optimal solutions | hyper planes | hyper planes | minimax theory | minimax theory | polyhedral convexity | polyhedral convexity | polyhedral cones | polyhedral cones | polyhedral sets | polyhedral sets | convex analysis | convex analysis | optimization | optimization | convexity | convexity | Lagrange multipliers | Lagrange multipliers | duality | duality | continuous optimization | continuous optimization | saddle point theory | saddle point theory | linear algebra | linear algebra | real analysis | real analysis | convex sets | convex sets | convex functions | convex functions | extreme points | extreme points | subgradients | subgradients | constrained optimization | constrained optimization | directional derivatives | directional derivatives | subdifferentials | subdifferentials | conical approximations | conical approximations | Fritz John optimality | Fritz John optimality | Exact penalty functions | Exact penalty functions | conjugate duality | conjugate duality | conjugate functions | conjugate functions | Fenchel duality | Fenchel duality | exact penalty functions | exact penalty functions | dual computational methods | dual computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.100A Introduction to Analysis (MIT) 18.100A Introduction to Analysis (MIT)

Description

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Subjects

mathematical analysis | mathematical analysis | estimations | estimations | limit of a sequence | limit of a sequence | limit theorems | limit theorems | subsequences | subsequences | cluster points | cluster points | infinite series | infinite series | power series | power series | local and global properties | local and global properties | continuity | continuity | intermediate-value theorem | intermediate-value theorem | convexity | convexity | integrability | integrability | Riemann integral | Riemann integral | calculus | calculus | convergence | convergence | Gamma function | Gamma function | Stirling | Stirling | quantifiers and negation | quantifiers and negation | Leibniz | Leibniz | Fubini | Fubini | improper integrals | improper integrals | Lebesgue integral | Lebesgue integral | mathematical proofs | mathematical proofs | differentiation | differentiation | integration | integration

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.253 Convex Analysis and Optimization (MIT) 6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood. This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood.

Subjects

convex analysis | convex analysis | convex optimization | convex optimization | hyperplanes | hyperplanes | conjugacy | conjugacy | polyhedral convexity | polyhedral convexity | geometric duality | geometric duality | duality theory | duality theory | subgradients | subgradients | optimality conditions | optimality conditions | convex optimization algorithms | convex optimization algorithms

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.253 Convex Analysis and Optimization (MIT)

Description

6.253 develops the core analytical issues of continuous optimization, duality, and saddle point theory, using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions is discussed in detail, and is the basis for an intuitive, highly visual, geometrical approach to the subject.

Subjects

affine hulls | recession cones | global minima | local minima | optimal solutions | hyper planes | minimax theory | polyhedral convexity | polyhedral cones | polyhedral sets | convex analysis | optimization | convexity | Lagrange multipliers | duality | continuous optimization | saddle point theory | linear algebra | real analysis | convex sets | convex functions | extreme points | subgradients | constrained optimization | directional derivatives | subdifferentials | conical approximations | Fritz John optimality | Exact penalty functions | conjugate duality | conjugate functions | Fenchel duality | exact penalty functions | dual computational methods

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.466 Mathematical Statistics (MIT) 18.466 Mathematical Statistics (MIT)

Description

This graduate level mathematics course covers decision theory, estimation, confidence intervals, and hypothesis testing. The course also introduces students to large sample theory. Other topics covered include asymptotic efficiency of estimates, exponential families, and sequential analysis. This graduate level mathematics course covers decision theory, estimation, confidence intervals, and hypothesis testing. The course also introduces students to large sample theory. Other topics covered include asymptotic efficiency of estimates, exponential families, and sequential analysis.

Subjects

decision theory | decision theory | testing simple hypothesis | testing simple hypothesis | Neyman-Pearson Lemma | Neyman-Pearson Lemma | Bayes decision theory | Bayes decision theory | sufficiency | sufficiency | estimation | estimation | convexity | convexity | Lehmann-Scheff'e property | Lehmann-Scheff'e property | Exponential families | Exponential families | Stein?s phenomenon | Stein?s phenomenon | James-Stein estimators | James-Stein estimators | M-estimation | M-estimation | Solari?s example | Solari?s example | Wilks?s theorem | Wilks?s theorem | stochastic processes | stochastic processes | probability theory | probability theory | confidence intervals | confidence intervals | large sample theory | large sample theory | sequential analysis | sequential analysis | asymptotic efficiency | asymptotic efficiency

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.253 Convex Analysis and Optimization (MIT) 6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a variety This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a variety

Subjects

convexity | convexity | optimization | optimization | geometric duality | geometric duality | Lagrangian duality | Lagrangian duality | Fenchel duality | Fenchel duality | cone programming | cone programming | semidefinite programming | semidefinite programming | subgradients | subgradients | constrained optimization | constrained optimization | gradient projection | gradient projection

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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6.972 Algebraic Techniques and Semidefinite Optimization (MIT) 6.972 Algebraic Techniques and Semidefinite Optimization (MIT)

Description

This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications. This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.

Subjects

algebraic and computational techniques | algebraic and computational techniques | optimization problems | optimization problems | polynomial equations | polynomial equations | inequalities | inequalities | semidefinite optimization | semidefinite optimization | convexity-based ideas | convexity-based ideas | complexity results | complexity results | efficient implementations | efficient implementations

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.466 Mathematical Statistics (MIT)

Description

This graduate level mathematics course covers decision theory, estimation, confidence intervals, and hypothesis testing. The course also introduces students to large sample theory. Other topics covered include asymptotic efficiency of estimates, exponential families, and sequential analysis.

Subjects

decision theory | testing simple hypothesis | Neyman-Pearson Lemma | Bayes decision theory | sufficiency | estimation | convexity | Lehmann-Scheff'e property | Exponential families | Stein?s phenomenon | James-Stein estimators | M-estimation | Solari?s example | Wilks?s theorem | stochastic processes | probability theory | confidence intervals | large sample theory | sequential analysis | asymptotic efficiency

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.100A Introduction to Analysis (MIT)

Description

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space. MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the pla

Subjects

mathematical analysis | estimations | limit of a sequence | limit theorems | subsequences | cluster points | infinite series | power series | local and global properties | continuity | intermediate-value theorem | convexity | integrability | Riemann integral | calculus | convergence | Gamma function | Stirling | quantifiers and negation | Leibniz | Fubini | improper integrals | Lebesgue integral | mathematical proofs | differentiation | integration

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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6.972 Algebraic Techniques and Semidefinite Optimization (MIT)

Description

This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.

Subjects

algebraic and computational techniques | optimization problems | polynomial equations | inequalities | semidefinite optimization | convexity-based ideas | complexity results | efficient implementations

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood.

Subjects

convex analysis | convex optimization | hyperplanes | conjugacy | polyhedral convexity | geometric duality | duality theory | subgradients | optimality conditions | convex optimization algorithms

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a variety

Subjects

convexity | optimization | geometric duality | Lagrangian duality | Fenchel duality | cone programming | semidefinite programming | subgradients | constrained optimization | gradient projection

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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