The Simplex Algorithm in Dimension Three1
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1 Lifts of Polytopes
Lecture 5: Lifts of polytopes and non-negative rank CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Lifts of polytopes 1.1 Polytopes and inequalities Recall that the convex hull of a subset X n is defined by ⊆ conv X λx + 1 λ x0 : x; x0 X; λ 0; 1 : ( ) f ( − ) 2 2 [ ]g A d-dimensional convex polytope P d is the convex hull of a finite set of points in d: ⊆ P conv x1;:::; xk (f g) d for some x1;:::; xk . 2 Every polytope has a dual representation: It is a closed and bounded set defined by a family of linear inequalities P x d : Ax 6 b f 2 g for some matrix A m d. 2 × Let us define a measure of complexity for P: Define γ P to be the smallest number m such that for some C s d ; y s ; A m d ; b m, we have ( ) 2 × 2 2 × 2 P x d : Cx y and Ax 6 b : f 2 g In other words, this is the minimum number of inequalities needed to describe P. If P is full- dimensional, then this is precisely the number of facets of P (a facet is a maximal proper face of P). Thinking of γ P as a measure of complexity makes sense from the point of view of optimization: Interior point( methods) can efficiently optimize linear functions over P (to arbitrary accuracy) in time that is polynomial in γ P . ( ) 1.2 Lifts of polytopes Many simple polytopes require a large number of inequalities to describe. -
Revised Primal Simplex Method Katta G
6.1 Revised Primal Simplex method Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor First put LP in standard form. This involves floowing steps. • If a variable has only a lower bound restriction, or only an upper bound restriction, replace it by the corresponding non- negative slack variable. • If a variable has both a lower bound and an upper bound restriction, transform lower bound to zero, and list upper bound restriction as a constraint (for this version of algorithm only. In bounded variable simplex method both lower and upper bound restrictions are treated as restrictions, and not as constraints). • Convert all inequality constraints as equations by introducing appropriate nonnegative slack for each. • If there are any unrestricted variables, eliminate each of them one by one by performing a pivot step. Each of these reduces no. of variables by one, and no. of constraints by one. This 128 is equivalent to having them as permanent basic variables in the tableau. • Write obj. in min form, and introduce it as bottom row of original tableau. • Make all RHS constants in remaining constraints nonnega- tive. 0 Example: Max z = x1 − x2 + x3 + x5 subject to x1 − x2 − x4 − x5 ≥ 2 x2 − x3 + x5 + x6 ≤ 11 x1 + x2 + x3 − x5 =14 −x1 + x4 =6 x1 ≥ 1;x2 ≤ 1; x3;x4 ≥ 0; x5;x6 unrestricted. 129 Revised Primal Simplex Algorithm With Explicit Basis Inverse. INPUT NEEDED: Problem in standard form, original tableau, and a primal feasible basic vector. Original Tableau x1 ::: xj ::: xn −z a11 ::: a1j ::: a1n 0 b1 . am1 ::: amj ::: amn 0 bm c1 ::: cj ::: cn 1 α Initial Setup: Let xB be primal feasible basic vector and Bm×m be associated basis. -
Simplicial Complexes
46 III Complexes III.1 Simplicial Complexes There are many ways to represent a topological space, one being a collection of simplices that are glued to each other in a structured manner. Such a collection can easily grow large but all its elements are simple. This is not so convenient for hand-calculations but close to ideal for computer implementations. In this book, we use simplicial complexes as the primary representation of topology. Rd k Simplices. Let u0; u1; : : : ; uk be points in . A point x = i=0 λiui is an affine combination of the ui if the λi sum to 1. The affine hull is the set of affine combinations. It is a k-plane if the k + 1 points are affinely Pindependent by which we mean that any two affine combinations, x = λiui and y = µiui, are the same iff λi = µi for all i. The k + 1 points are affinely independent iff P d P the k vectors ui − u0, for 1 ≤ i ≤ k, are linearly independent. In R we can have at most d linearly independent vectors and therefore at most d+1 affinely independent points. An affine combination x = λiui is a convex combination if all λi are non- negative. The convex hull is the set of convex combinations. A k-simplex is the P convex hull of k + 1 affinely independent points, σ = conv fu0; u1; : : : ; ukg. We sometimes say the ui span σ. Its dimension is dim σ = k. We use special names of the first few dimensions, vertex for 0-simplex, edge for 1-simplex, triangle for 2-simplex, and tetrahedron for 3-simplex; see Figure III.1. -
Degrees of Freedom in Quadratic Goodness of Fit
Submitted to the Annals of Statistics DEGREES OF FREEDOM IN QUADRATIC GOODNESS OF FIT By Bruce G. Lindsay∗, Marianthi Markatouy and Surajit Ray Pennsylvania State University, Columbia University, Boston University We study the effect of degrees of freedom on the level and power of quadratic distance based tests. The concept of an eigendepth index is introduced and discussed in the context of selecting the optimal de- grees of freedom, where optimality refers to high power. We introduce the class of diffusion kernels by the properties we seek these kernels to have and give a method for constructing them by exponentiating the rate matrix of a Markov chain. Product kernels and their spectral decomposition are discussed and shown useful for high dimensional data problems. 1. Introduction. Lindsay et al. (2008) developed a general theory for good- ness of fit testing based on quadratic distances. This class of tests is enormous, encompassing many of the tests found in the literature. It includes tests based on characteristic functions, density estimation, and the chi-squared tests, as well as providing quadratic approximations to many other tests, such as those based on likelihood ratios. The flexibility of the methodology is particularly important for enabling statisticians to readily construct tests for model fit in higher dimensions and in more complex data. ∗Supported by NSF grant DMS-04-05637 ySupported by NSF grant DMS-05-04957 AMS 2000 subject classifications: Primary 62F99, 62F03; secondary 62H15, 62F05 Keywords and phrases: Degrees of freedom, eigendepth, high dimensional goodness of fit, Markov diffusion kernels, quadratic distance, spectral decomposition in high dimensions 1 2 LINDSAY ET AL. -
Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order. -
THE DIMENSION of a VECTOR SPACE 1. Introduction This Handout
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD 1. Introduction This handout is a supplementary discussion leading up to the definition of dimension of a vector space and some of its properties. We start by defining the span of a finite set of vectors and linear independence of a finite set of vectors, which are combined to define the all-important concept of a basis. Definition 1.1. Let V be a vector space over a field F . For any finite subset fv1; : : : ; vng of V , its span is the set of all of its linear combinations: Span(v1; : : : ; vn) = fc1v1 + ··· + cnvn : ci 2 F g: Example 1.2. In F 3, Span((1; 0; 0); (0; 1; 0)) is the xy-plane in F 3. Example 1.3. If v is a single vector in V then Span(v) = fcv : c 2 F g = F v is the set of scalar multiples of v, which for nonzero v should be thought of geometrically as a line (through the origin, since it includes 0 · v = 0). Since sums of linear combinations are linear combinations and the scalar multiple of a linear combination is a linear combination, Span(v1; : : : ; vn) is a subspace of V . It may not be all of V , of course. Definition 1.4. If fv1; : : : ; vng satisfies Span(fv1; : : : ; vng) = V , that is, if every vector in V is a linear combination from fv1; : : : ; vng, then we say this set spans V or it is a spanning set for V . Example 1.5. In F 2, the set f(1; 0); (0; 1); (1; 1)g is a spanning set of F 2. -
Chapter 6: Gradient-Free Optimization
AA222: MDO 133 Monday 30th April, 2012 at 16:25 Chapter 6 Gradient-Free Optimization 6.1 Introduction Using optimization in the solution of practical applications we often encounter one or more of the following challenges: • non-differentiable functions and/or constraints • disconnected and/or non-convex feasible space • discrete feasible space • mixed variables (discrete, continuous, permutation) • large dimensionality • multiple local minima (multi-modal) • multiple objectives Gradient-based optimizers are efficient at finding local minima for high-dimensional, nonlinearly- constrained, convex problems; however, most gradient-based optimizers have problems dealing with noisy and discontinuous functions, and they are not designed to handle multi-modal problems or discrete and mixed discrete-continuous design variables. Consider, for example, the Griewank function: n n x2 f(x) = P i − Q cos pxi + 1 4000 i i=1 i=1 (6.1) −600 ≤ xi ≤ 600 Mixed (Integer-Continuous) Figure 6.1: Graphs illustrating the various types of functions that are problematic for gradient- based optimization algorithms AA222: MDO 134 Monday 30th April, 2012 at 16:25 Figure 6.2: The Griewank function looks deceptively smooth when plotted in a large domain (left), but when you zoom in, you can see that the design space has multiple local minima (center) although the function is still smooth (right) How we could find the best solution for this example? • Multiple point restarts of gradient (local) based optimizer • Systematically search the design space • Use gradient-free optimizers Many gradient-free methods mimic mechanisms observed in nature or use heuristics. Unlike gradient-based methods in a convex search space, gradient-free methods are not necessarily guar- anteed to find the true global optimal solutions, but they are able to find many good solutions (the mathematician's answer vs. -
Linear Algebra Handout
Artificial Intelligence: 6.034 Massachusetts Institute of Technology April 20, 2012 Spring 2012 Recitation 10 Linear Algebra Review • A vector is an ordered list of values. It is often denoted using angle brackets: ha; bi, and its variable name is often written in bold (z) or with an arrow (~z). We can refer to an individual element of a vector using its index: for example, the first element of z would be z1 (or z0, depending on how we're indexing). Each element of a vector generally corresponds to a particular dimension or feature, which could be discrete or continuous; often you can think of a vector as a point in Euclidean space. p 2 2 2 • The magnitude (also called norm) of a vector x = hx1; x2; :::; xni is x1 + x2 + ::: + xn, and is denoted jxj or kxk. • The sum of a set of vectors is their elementwise sum: for example, ha; bi + hc; di = ha + c; b + di (so vectors can only be added if they are the same length). The dot product (also called scalar product) of two vectors is the sum of their elementwise products: for example, ha; bi · hc; di = ac + bd. The dot product x · y is also equal to kxkkyk cos θ, where θ is the angle between x and y. • A matrix is a generalization of a vector: instead of having just one row or one column, it can have m rows and n columns. A square matrix is one that has the same number of rows as columns. A matrix's variable name is generally a capital letter, often written in bold. -
11.1 Algorithms for Linear Programming
6.854 Advanced Algorithms Lecture 11: 10/18/2004 Lecturer: David Karger Scribes: David Schultz 11.1 Algorithms for Linear Programming 11.1.1 Last Time: The Ellipsoid Algorithm Last time, we touched on the ellipsoid method, which was the first polynomial-time algorithm for linear programming. A neat property of the ellipsoid method is that you don’t have to be able to write down all of the constraints in a polynomial amount of space in order to use it. You only need one violated constraint in every iteration, so the algorithm still works if you only have a separation oracle that gives you a separating hyperplane in a polynomial amount of time. This makes the ellipsoid algorithm powerful for theoretical purposes, but isn’t so great in practice; when you work out the details of the implementation, the running time winds up being O(n6 log nU). 11.1.2 Interior Point Algorithms We will finish our discussion of algorithms for linear programming with a class of polynomial-time algorithms known as interior point algorithms. These have begun to be used in practice recently; some of the available LP solvers now allow you to use them instead of Simplex. The idea behind interior point algorithms is to avoid turning corners, since this was what led to combinatorial complexity in the Simplex algorithm. They do this by staying in the interior of the polytope, rather than walking along its edges. A given constrained linear optimization problem is transformed into an unconstrained gradient descent problem. To avoid venturing outside of the polytope when descending the gradient, these algorithms use a potential function that is small at the optimal value and huge outside of the feasible region. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Chapter 1 Introduction
Chapter 1 Introduction 1.1 Motivation The simplex method was invented by George B. Dantzig in 1947 for solving linear programming problems. Although, this method is still a viable tool for solving linear programming problems and has undergone substantial revisions and sophistication in implementation, it is still confronted with the exponential worst-case computational effort. The question of the existence of a polynomial time algorithm was answered affirmatively by Khachian [16] in 1979. While the theoretical aspects of this procedure were appealing, computational effort illustrated that the new approach was not going to be a competitive alternative to the simplex method. However, when Karmarkar [15] introduced a new low-order polynomial time interior point algorithm for solving linear programs in 1984, and demonstrated this method to be computationally superior to the simplex algorithm for large, sparse problems, an explosion of research effort in this area took place. Today, a variety of highly effective variants of interior point algorithms exist (see Terlaky [32]). On the other hand, while exterior penalty function approaches have been widely used in the context of nonlinear programming, their application to solve linear programming problems has been comparatively less explored. The motivation of this research effort is to study how several variants of exterior penalty function methods, suitably modified and fine-tuned, perform in comparison with each other, and to provide insights into their viability for solving linear programming problems. Many experimental investigations have been conducted for studying in detail the simplex method and several interior point algorithms. Whereas these methods generally perform well and are sufficiently adequate in practice, it has been demonstrated that solving linear programming problems using a simplex based algorithm, or even an interior-point type of procedure, can be inadequately slow in the presence of complicating constraints, dense coefficient matrices, and ill- conditioning. -
The Number of Degrees of Freedom of a Gauge Theory (Addendum to the Discussion on Pp
The number of degrees of freedom of a gauge theory (Addendum to the discussion on pp. 59–62 of notes) Let us work in the Fourier picture (Remark, p. 61 of notes). In a general gauge, the Maxwell equation for the vector potential is µ α α µ α −∂ ∂µA + ∂ ∂µA = J . (1) Upon taking Fourier transforms, this becomes µ α α µ α ′ k kµA − k kµA = J , (1 ) where A~ and J~ are now functions of the 4-vector ~k. (One would normally denote the transforms by a caret (Aˆα, etc.), but for convenience I won’t.) The field strength tensor is F αβ = ∂αAβ − ∂βAα, (2) or F αβ = ikαAβ − ikβAα. (2′) The relation between field and current is (factor 4π suppressed) α αβ J = ∂βF , (3) or α αµ ′ J = ikµF . (3 ) Of course, (2′) and (3′) imply (1′). (1′) can be written in matrix form as J~ = MA,~ (4) 2 0 0 0 0 ~k − k k0 −k k1 −k k2 −k k3 1 ~ 2 1 1 1 ~ µ ~ −k k0 k − k k1 −k k2 −k k3 M(k) = k kµ I − k ⊗ k˜ = 2 2 2 2 2 (5) −k k0 −k k1 ~k − k k2 −k k3 3 3 3 2 3 −k k0 −k k1 −k k2 ~k − k k3 Consider a generic ~k (not a null vector). Suppose that A~ is a multiple of ~k : Aa(~k) = kαχ(~k). (6′) Then it is easy to see that A~ is in the kernel (null space) of M(~k); that is, it yields J~(~k) = 0.