Category: Materialismo Dialéctico

Robert Lucas has died at the age of 85.  Lucas was a leading mainstream neoclassical economist at the University of Chicago – the bastion of neoclassical equilibrium economic theory.  In 1995, Lucas received a ‘Nobel prize’ for his theory of ‘rational expectations’.  He was regarded by Greg Mankiw, the author of the main mainstream economics […]

Robert Lucas: the rationality of capitalism

In his latest book, FT columnist and Keynesian guru Martin Wolf, starts from the premise that capitalism and democracy go together like a hand in a glove.  But he is worried.  “We are living in an age when economic failings have shaken faith in global capitalism. Some now argue that capitalism is better without democracy; […]

The crisis of democratic capitalism

Rachel Greenfeld and I have just uploaded to the arXiv our announcement “A counterexample to the periodic tiling conjecture“. This is an announcement of a longer paper that we are currently in the process of writing up (and hope to release in a few weeks), in which we disprove the periodic tiling conjecture of Grünbaum-Shephard […]

A counterexample to the periodic tiling conjecture

This post clears up a confusion I ran into recently with directional semi-derivatives in one dimension. Directional semi-derivatives Consider a function , where , and let be a direction in (i.e. a vector with length 1). For any point , the directional semi-derivative of at in the direction is given by if the limit exists. […]

Clearing up directional semi-derivatives for one dimension

Let be a set of points in , and let denote the convex hull of . We are often interested in finding the point in that is nearest to some reference point , known as the Euclidean projection onto . Let’s denote this point by . Optimality condition for when Wolfe (1976) (Reference 1) provides […]

Optimality condition for Euclidean projection onto a convex polytope

Duchi et. al. (2008) (Reference 1) has a simple and efficient algorithm for projecting a point onto the simplex. Formally, the optimization problem is where represents the Euclidean norm and is some constant. Their algorithm runs in time, with the most expensive step being sorting the elements of in descending order. Here is the algorithm: […]

O(n log n) algorithm for Euclidean projection onto a simplex

Relations and resolvents A relation on is a subset of . We use the notation to mean the set . You can think of as an operator that maps vectors to sets . (Along this line of thinking, functions are a special kind of relation where every vector is mapped to a set consisting of […]

Relations, resolvents and the proximal operator

Let be a convex function. Moreau’s Decomposition Theorem is the following result: Theorem (Moreau Decomposition). For all , , where is the proximal operator for and is the convex conjugate of . Here is the proof: Let . Then, The second equivalence is a result involving convex conjugates and subdifferentials (see this post for statement […]

What is Moreau’s Decomposition Theorem?

Convex conjugate functions and Fenchel’s inequality Let be some function. The convex conjugate of (also know as the Fenchel transformation), is the function defined as Fenchel’s equality is the following statement: Theorem (Fenchel’s inequality). For any , The proof of Fenchel’s equality follows directly from the definition of the convex conjugate: A direct application of […]

Convex conjugate functions, Fenchel’s inequality, subdifferentials and optimality