watchmykidspc.com

convexoptimization.com

Convex Optimization - Duality Gap

http://www.convexoptimization.com/dattorro/duality.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. Essentially, duality theory concerns representation of a given optimization problem as half a. Given any real function f(x,z). Minimize x maximize z. X,z) = maximize z minimize x. Always holds. When. Minimize x maximize z. X,z) = maximize z minimize x. Exists, as depicted, and the. G(z) kisses f(x) ).

convexoptimization.com

Convex Optimization - Convex Geometry

http://www.convexoptimization.com/dattorro/convex_geometry.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. The most fundamental principle in convex geometry. Follows from the geometric Hahn-Banach theorem which guarantees any closed convex set to be an intersection of halfspaces. The second most fundamental principle of convex geometry also follows from the geometric Hahn-Banach theorem that guarantees existence of at least one hyperplane supporting a convex set (having nonempty interior) at each point on its boundary.

meboo.convexoptimization.com

Convex Optimization - Products from Meboo Publishing

http://meboo.convexoptimization.com/gallery.html

From too much study, and from extreme passion, cometh madnesse. —Isaac Newton.

convexoptimization.com

Convex Optimization - Convex Cones

http://www.convexoptimization.com/dattorro/convex_cones.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. We call a set. Iff any nonnegative combination of elements from. The set of all convex cones is a proper subset of all cones. The set of convex cones is a narrower but more familiar class of cone, any member of which can be equivalently described as the intersection of a possibly (but not necessarily) infinite number of hyperplanes (. Through the origin ). And halfspaces whose bounding hyperplanes pass through the origin; a. And any polyhedral cone;.

convexoptimization.com

Convex Optimization - Calculus of Inequalities

http://www.convexoptimization.com/dattorro/calculus_of_inequalities.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. Convex Analysis is the calculus of inequalities while Convex Optimization is its application. For example, myriad alternative systems of linear inequality can be explained in terms of pointed closed convex cones and their duals.

convexoptimization.com

Convex Optimization - Conic Independence

http://www.convexoptimization.com/dattorro/conic_independence.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. Conic independence is introduced as a natural extension to linear and affine independence; a new tool in convex analysis most useful for manipulation of cones. Perhaps the most useful application of conic independence is determination of the intersection of closed convex cones from their halfspace-descriptions, or representation of the sum of closed convex cones from their vertex-descriptions.

convexoptimization.com

Convex Optimization - Elliptope and Fantope

http://www.convexoptimization.com/dattorro/elliptope_and_fantope.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. The early proponents of this convex body defined. Above is that pillow-shaped elliptope formed from all positive semidefinite 3x3 matrices having 1 along the main diagonal. In the example illustrated above, the elliptope is that line segment interior to the positive semidefinite cone of 2x2 matrices. Is our nomenclature named after mathematician Ky Fan. In the example illustrated, the circular Fantope represents outer product of all 2x2 rank-1 orthonormal mat...

convexoptimization.com

Convex Optimization - Convex Functions

http://www.convexoptimization.com/dattorro/convex_functions.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. The link between convex sets and convex functions. Is via the epigraph: A function is convex if and only if its epigraph is a convex set.". Any convex real function f(X) has unique minimum value over any convex subset of its domain. Yet solution to some convex optimization problem is, in general, not unique;. For the optimal solution set to be a single point, it is sufficient that f(X) be a strictly convex real function and set C convex.

convexoptimization.com

Convex Optimization - Convex Optimization

http://www.convexoptimization.com/dattorro/convex_optimization.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. Prior to 1984 [renaissance of interior-point methods of solution] linear and nonlinear programming, one a subset of the other, had evolved for the most part along unconnected paths, without even a common terminology. The use of. Serves as a persistent reminder of these differences.". The goal, then, becomes conversion of a given problem (. Perhaps a nonconvex problem statement ). A subset of dom g ). Subject to X in C. Where constraints are abstract here in t...

convexoptimization.com

Convex Optimization - Distance Geometry

http://www.convexoptimization.com/dattorro/distance_geometry.html

Convex, Affine, Conic: Hulls. Harold W. Kuhn. If we agree that a set of points can have a shape three points can form a triangle and its interior, for example, four points a tetrahedron), then we can ascribe shape of a set of points to their convex hull. It should be apparent: these shapes can be determined only to within a rigid transformation (a rotation, reflection, and a translation).