Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Proof of Strong Duality

In course

MATH-329: Continuous optimization

This course introduces students to continuous, nonlinear optimization. We study the theory of optimization with continuous variables (with full proofs), and we analyze and implement important algorithms to solve constrained and unconstrained problems. ...

Description

This lecture covers the proof of strong duality in optimization problems, demonstrating the relationship between primal and dual problems, the existence of valid Lagrange multipliers, and the convexity of the objective function. Examples of Rayleigh quotient optimization are provided to illustrate the concepts discussed.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Nicolas Boumal

Official source

Related concepts (32)

Duality (optimization)

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem.

Mathematical proof

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".

Duality (mathematics)

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.

Proof theory

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

Proof (truth)

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. The concept applies in a variety of disciplines, with both the nature of the evidence or justification and the criteria for sufficiency being area-dependent. In the area of oral and written communication such as conversation, dialog, rhetoric, etc., a proof is a persuasive perlocutionary speech act, which demonstrates the truth of a proposition.

Related lectures (54)

Approximation Algorithms

Covers approximation algorithms for optimization problems, LP relaxation, and randomized rounding techniques.

Optimization with Constraints: KKT Conditions

Covers the KKT conditions for optimization with constraints, essential for solving constrained optimization problems efficiently.

Linear Programming Duality

Explores linear programming duality, covering constraints, variables, solutions, and the relationship between primal and dual LP.

Duality: Duality in Linear Optimization MOOC: Optimization: principles and algorithms - Linear optimization

Covers the concept of linear optimization and the duality relationship between primal and dual problems.

Optimization Duality: Theory and Algorithms