Fundamentals of Optimization.

Explore 27 formal proofs and analytical renders within the discipline of Fundamentals of Optimization.

Intermediate

The Lagrange Multiplier Theorem (First-Order Necessary Conditions for Equality Constraints)

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Advanced

The Karush-Kuhn-Tucker (KKT) Conditions: General First-Order Necessary Conditions for Constrained Optimization

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Advanced

Optimality of Interval Reduction in the Golden Section Search Method

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Advanced

Optimality of Interval Reduction in the Fibonacci Search Method

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Advanced

Proof of Affine Invariance for Newton's Method

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Intermediate

The Descent Property of Gradient-Based Optimization Methods

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Advanced

Slater's Condition and Strong Duality for Convex Nonlinear Programs

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Advanced

The Envelope Theorem: Sensitivity of the Optimal Value Function

Exploring the cinematic intuition of The Envelope Theorem: Sensitivity of the Optimal Value Function.

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Advanced

Local Quadratic Convergence of Sequential Quadratic Programming (SQP)

Exploring the cinematic intuition of Local Quadratic Convergence of Sequential Quadratic Programming (SQP).

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Foundational

Proof that the Epigraph of a Convex Function is a Convex Set

Exploring the cinematic intuition of Proof that the Epigraph of a Convex Function is a Convex Set.

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Weierstrass Extreme Value Theorem: Guaranteeing Existence of Optima

Local Optima are Global Optima for Convex Functions

Hessian Matrix and Second-Order Optimality Conditions

Jensen's Inequality for Convex Functions

Proof that the Intersection of Convex Sets is Convex

The Hyperplane Separation Theorem for Convex Sets

Farkas' Lemma: A Fundamental Theorem of Alternatives

Properties of Subgradients for Non-Differentiable Convex Functions

Convexity of the Feasible Region in Linear Programming

The Fundamental Theorem of Linear Programming: Optimality at Extreme Points

The Strong Duality Theorem for Linear Programming

Simplex Method Termination and Bland's Rule

Convergence Proof of the Steepest Descent Method

Proof of Quadratic Convergence for Newton's Method

Finite Termination of Conjugate Gradient Method for Quadratic Functions

Orthogonality of Conjugate Directions

Proof of Superlinear Convergence for the Secant Method

The Lagrange Multiplier Theorem (First-Order Necessary Conditions for Equality Constraints)

The Karush-Kuhn-Tucker (KKT) Conditions: General First-Order Necessary Conditions for Constrained Optimization

Optimality of Interval Reduction in the Golden Section Search Method

Optimality of Interval Reduction in the Fibonacci Search Method

Proof of Affine Invariance for Newton's Method

The Descent Property of Gradient-Based Optimization Methods

Slater's Condition and Strong Duality for Convex Nonlinear Programs

The Envelope Theorem: Sensitivity of the Optimal Value Function

Local Quadratic Convergence of Sequential Quadratic Programming (SQP)

Proof that the Epigraph of a Convex Function is a Convex Set