**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Linear Constraints and Vertices

Description

This lecture focuses on the importance of vertices in optimization, explaining how to identify them in a polyhedron by looking at convex combinations of feasible points. The instructor presents a theorem stating that a polyhedron in standard form contains at least one vertex. The process of finding vertices involves activating constraints and using linear algebra to ensure feasibility. By setting non-basic variables to zero and checking the basic variables' positivity, one can efficiently determine the vertices of the constraint polyhedron, a crucial concept in optimization.

Login to watch the video

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In MOOCs (6)

Related concepts (16)

Instructor

Optimization: principles and algorithms - Linear optimization

Introduction to linear optimization, duality and the simplex algorithm.

Optimization: principles and algorithms - Linear optimization

Introduction to linear optimization, duality and the simplex algorithm.

Optimization: principles and algorithms - Network and discrete optimization

Introduction to network optimization and discrete optimization

Optimization: principles and algorithms - Network and discrete optimization

Introduction to network optimization and discrete optimization

Optimization: principles and algorithms - Unconstrained nonlinear optimization

Introduction to unconstrained nonlinear optimization, Newton’s algorithms and descent methods.

Polyhedron

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions.

Dual polyhedron

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron. Duality preserves the symmetries of a polyhedron.

Flexible polyhedron

In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions). The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces isometric to an octahedron.

Schönhardt polyhedron

In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who described it in 1928. The same polyhedra have also been studied in connection with Cauchy's rigidity theorem as an example where polyhedra with two different shapes have faces of the same shapes. One way of constructing the Schönhardt polyhedron starts with a triangular prism, with two parallel equilateral triangles as its faces.

Császár polyhedron

In geometry, the Császár polyhedron (ˈt͡ʃaːsaːr) is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K_7 onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.