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Stochastic semidefinite programs (SSDP's) are a new class of optimization problems with a wide variety applications proposed by the PI and his doctoral students. The broad objective of this project was to develop applications of and algorithms for SSDP's. We have developed five classes of novel applications, and three classes of new algorithms. We have proved the convergence and polynomial complexity of the algorithms. We have also identified two new classes of optimization problems which may be useful for future research.

This report results from a contract tasking ILOG as follows: Several promising areas will be investigated: more efficient use of linear programming. new and more powerful relaxation techniques use of column generation techniques integration of graph theoretic algorithms use of randomization to search branch-and-bound trees and symmetry breaking. Techniques that are developed will be tested on challenging problems where current constraint programming technology is limited.

This report results from a contract tasking ILOG as follows: Several promising areas will be investigated: more efficient use of linear programming. new and more powerful relaxation techniques use of column generation techniques integration of graph theoretic algorithms use of randomization to search branch-and-bound trees and symmetry breaking. Techniques that are developed will be tested on challenging problems where current constraint programming technology is limited.

ai how i fail

884 pages : 28 cm

THIS TUTORIAL IS SOLD OUT . The session presents many of the techniques that leading games are using. It covers the most important local and global lighting models, multi-frustum shadow mapping techniques offering high resolution shadows, methods for efficient compression of geometry, and error-tolerant calculation of triangle mesh tangent space. Focusing on the wider engine scope are sections on the design of an efficient particle system, the implementation of a post-processing pipeline and an overview of the pitfalls and challenges of integrating advanced surface shaders into a game engine. Finally, because shaders need to run on several different hardware platforms and graphics cards, there is a session covering tips and tricks for multi-platform optimizations. 1. Lighting a Game Scene (1 hour) a. Local Illumination by Matthias Wloka b. Global Illumination by Carsten Dachsbacher 2. Shadow Maps by Tom Forsyth (1 hour) 3. Manipulating Geometry (1.5 hour) a. Using Vertex Shaders for Geometry Compression by Kenneth L. Hurley b. Triangle Mesh Tangent Space Calcuation by Martin Mittring c. Real-Time Deformation Methods by Vlad Stamate 4. A Particle Effect System by Lutz Latta (1 hour) 5. Post-Processing Pipeline by Wolfgang Engel (1 hour) 6. Integrating Shaders into a Game Engine by Tom Forsyth (30 min.) 7. Performance Optimizations by Matthias Wloka (30 min.)

THIS TUTORIAL IS SOLD OUT . The session presents many of the techniques that leading games are using. It covers the most important local and global lighting models, multi-frustum shadow mapping techniques offering high resolution shadows, methods for efficient compression of geometry, and error-tolerant calculation of triangle mesh tangent space. Focusing on the wider engine scope are sections on the design of an efficient particle system, the implementation of a post-processing pipeline and an overview of the pitfalls and challenges of integrating advanced surface shaders into a game engine. Finally, because shaders need to run on several different hardware platforms and graphics cards, there is a session covering tips and tricks for multi-platform optimizations. 1. Lighting a Game Scene (1 hour) a. Local Illumination by Matthias Wloka b. Global Illumination by Carsten Dachsbacher 2. Shadow Maps by Tom Forsyth (1 hour) 3. Manipulating Geometry (1.5 hour) a. Using Vertex Shaders for Geometry Compression by Kenneth L. Hurley b. Triangle Mesh Tangent Space Calcuation by Martin Mittring c. Real-Time Deformation Methods by Vlad Stamate 4. A Particle Effect System by Lutz Latta (1 hour) 5. Post-Processing Pipeline by Wolfgang Engel (1 hour) 6. Integrating Shaders into a Game Engine by Tom Forsyth (30 min.) 7. Performance Optimizations by Matthias Wloka (30 min.)

Outline: 1) Introduction * Optimal control * Dynamic programming (DP). 2) Path Planning * Discrete planning as optimal control * Dijkstra's algorithm & its problems * Continuous DP & the Hamilton-Jacobi (HJ) PDE * The fast marching method (FMM): Dijkstra's for continuous spaces. 3) Algorithms for Static HJ PDEs * Four alternatives * FMM pros & cons. 4) Generalizations * Alternative action norms * Multiple objective planning.

Outline: 1) Introduction * Optimal control * Dynamic programming (DP). 2) Path Planning * Discrete planning as optimal control * Dijkstra's algorithm & its problems * Continuous DP & the Hamilton-Jacobi (HJ) PDE * The fast marching method (FMM): Dijkstra's for continuous spaces. 3) Algorithms for Static HJ PDEs * Four alternatives * FMM pros & cons. 4) Generalizations * Alternative action norms * Multiple objective planning.

vi, 312 pages : 28 cm

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax=b such that mXn matrix A is sparse and the graph, G = (V,E), of the matrix H. Our algorithm uses O(log(m+n) log squared s(m+n)) steps and or = s cubed (m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems. The objective of this paper is to reexamine the time-complexity of the l.l.s.p. and to indicate the possibility of speeding up its solution using the parallel algorithms of PR in brackets. As a major consequence, (which may become decisive for determining the best algorithm for the linear programming problem (l.p.p) at least over some important classes of instances of that problem), we will substantially speed up Karmarkar's algorithm, K, for the l.p.p. because solving the l.l.s.p. constitutes the most costly part of every iteration of that algorithm. A very wide range of applications leads to in particular the acceleration of the simplex algorithms, see C,M, for sparse l.p. problems. Further applications may include several combinatorial computations. In the next section we recall two known representations of the l.l.s.p. using normal equations. In sect. 3 we reexamine the computational cost of sequential algorithms for l.l.s.p. In sect. 4 we estimate the cost of performing our parallel algorithm for the same problem. In sect. 5 we consider one of the major applications of our results, that is, to the acceleration of Karmarkar's algorithm. In Appendix we will briefly comment on the current estimates for the computational cost of solving the l.p.p. Keywords: Nested dissection; Linear programming; Karmarkar's algorithm.

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax=b such that mXn matrix A is sparse and the graph, G = (V,E), of the matrix H. Our algorithm uses O(log(m+n) log squared s(m+n)) steps and or = s cubed (m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems. The objective of this paper is to reexamine the time-complexity of the l.l.s.p. and to indicate the possibility of speeding up its solution using the parallel algorithms of PR in brackets. As a major consequence, (which may become decisive for determining the best algorithm for the linear programming problem (l.p.p) at least over some important classes of instances of that problem), we will substantially speed up Karmarkar's algorithm, K, for the l.p.p. because solving the l.l.s.p. constitutes the most costly part of every iteration of that algorithm. A very wide range of applications leads to in particular the acceleration of the simplex algorithms, see C,M, for sparse l.p. problems. Further applications may include several combinatorial computations. In the next section we recall two known representations of the l.l.s.p. using normal equations. In sect. 3 we reexamine the computational cost of sequential algorithms for l.l.s.p. In sect. 4 we estimate the cost of performing our parallel algorithm for the same problem. In sect. 5 we consider one of the major applications of our results, that is, to the acceleration of Karmarkar's algorithm. In Appendix we will briefly comment on the current estimates for the computational cost of solving the l.p.p. Keywords: Nested dissection; Linear programming; Karmarkar's algorithm.

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax=b such that mXn matrix A is sparse and the graph, G = (V,E), of the matrix H. Our algorithm uses O(log(m+n) log squared s(m+n)) steps and or = s cubed (m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems. The objective of this paper is to reexamine the time-complexity of the l.l.s.p. and to indicate the possibility of speeding up its solution using the parallel algorithms of PR in brackets. As a major consequence, (which may become decisive for determining the best algorithm for the linear programming problem (l.p.p) at least over some important classes of instances of that problem), we will substantially speed up Karmarkar's algorithm, K, for the l.p.p. because solving the l.l.s.p. constitutes the most costly part of every iteration of that algorithm. A very wide range of applications leads to in particular the acceleration of the simplex algorithms, see C,M, for sparse l.p. problems. Further applications may include several combinatorial computations. In the next section we recall two known representations of the l.l.s.p. using normal equations. In sect. 3 we reexamine the computational cost of sequential algorithms for l.l.s.p. In sect. 4 we estimate the cost of performing our parallel algorithm for the same problem. In sect. 5 we consider one of the major applications of our results, that is, to the acceleration of Karmarkar's algorithm. In Appendix we will briefly comment on the current estimates for the computational cost of solving the l.p.p. Keywords: Nested dissection; Linear programming; Karmarkar's algorithm.

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax=b such that mXn matrix A is sparse and the graph, G = (V,E), of the matrix H. Our algorithm uses O(log(m+n) log squared s(m+n)) steps and or = s cubed (m+n) processors; it relies on our recent parallel algorithm for solving sparse linear systems. The objective of this paper is to reexamine the time-complexity of the l.l.s.p. and to indicate the possibility of speeding up its solution using the parallel algorithms of PR in brackets. As a major consequence, (which may become decisive for determining the best algorithm for the linear programming problem (l.p.p) at least over some important classes of instances of that problem), we will substantially speed up Karmarkar's algorithm, K, for the l.p.p. because solving the l.l.s.p. constitutes the most costly part of every iteration of that algorithm. A very wide range of applications leads to in particular the acceleration of the simplex algorithms, see C,M, for sparse l.p. problems. Further applications may include several combinatorial computations. In the next section we recall two known representations of the l.l.s.p. using normal equations. In sect. 3 we reexamine the computational cost of sequential algorithms for l.l.s.p. In sect. 4 we estimate the cost of performing our parallel algorithm for the same problem. In sect. 5 we consider one of the major applications of our results, that is, to the acceleration of Karmarkar's algorithm. In Appendix we will briefly comment on the current estimates for the computational cost of solving the l.p.p. Keywords: Nested dissection; Linear programming; Karmarkar's algorithm.

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

A nonlinear 0-1 program can be restated as a multilinear 0-1 program, which in turn is known to be equivalent to a linear 0-1 program with generalized covering (g.c.) inequalities. In a companion paper 6 we have defined a family of linear inequalities that contains more compact (smaller cardinality) linearizations of a multilinear 0-1 program than the one based on the g.c. inequalities. In this paper we analyze the dominance relations between inequalities of the above family. In particular, we give a criterion that can be checked in linear time, for deciding whether a g.c. inequality can be strengthened by extending the cover from which it was derived. We then describe a class of algorithms based on these results and discuss our computational experience. We conclude that the g.c. inequalities can be strengthened most of the time an extent that increases with problem density. In particular, the algorithm using the strengthening procedure outperforms the one using only g.c. inequalities whenever the number of nonlinear terms per constraint exceeds about 12-15, and the difference in their performance grows with the number of such terms. (Author)

http://uf.catalog.fcla.edu/uf.jsp?st=UF024909362&ix=pm&I=0&V=D&pm=1

http://uf.catalog.fcla.edu/uf.jsp?st=UF024909362&ix=pm&I=0&V=D&pm=1

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

A number of algorithms were developed for image warping and pyramid image filtering. Techniques were investigated for the parallel processing of a large number of independent irregular shaped regions on the MPP. In addition some utilities for dealing with very long vectors and for sorting were developed. Documentation pages for the algorithms which are available for distribution are given. The performance of the MPP for a number of basic data manipulations was determined. From these results it is possible to predict the efficiency of the MPP for a number of algorithms and applications. The Parallel Pascal development system, which is a portable programming environment for the MPP, was improved and better documentation including a tutorial was written. This environment allows programs for the MPP to be developed on any conventional computer system; it consists of a set of system programs and a library of general purpose Parallel Pascal functions. The algorithms were tested on the MPP and a presentation on the development system was made to the MPP users group. The UNIX version of the Parallel Pascal System was distributed to a number of new sites.

A number of algorithms were developed for image warping and pyramid image filtering. Techniques were investigated for the parallel processing of a large number of independent irregular shaped regions on the MPP. In addition some utilities for dealing with very long vectors and for sorting were developed. Documentation pages for the algorithms which are available for distribution are given. The performance of the MPP for a number of basic data manipulations was determined. From these results it is possible to predict the efficiency of the MPP for a number of algorithms and applications. The Parallel Pascal development system, which is a portable programming environment for the MPP, was improved and better documentation including a tutorial was written. This environment allows programs for the MPP to be developed on any conventional computer system; it consists of a set of system programs and a library of general purpose Parallel Pascal functions. The algorithms were tested on the MPP and a presentation on the development system was made to the MPP users group. The UNIX version of the Parallel Pascal System was distributed to a number of new sites.

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

Donald Knuth is one of the greatest and most impactful computer scientists and mathematicians ever. He is the recipient in 1974 of the Turing Award, considered the Nobel Prize of computing. He is the author of the multi-volume work, the magnum opus, The Art of Computer Programming. He made several key contributions to the rigorous analysis of the computational complexity of algorithms. He popularized asymptotic notation, that we all affectionately know as the big-O notation. He also created the TeX typesetting which most computer scientists, physicists, mathematicians, and scientists and engineers use to write technical papers and make them look

We consider the 0-1 Penalized Knapsack Problem (PKP). Each item has a profit, a weight and a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the items included in a solution. We propose an exact approach relying on a procedure which narrows the relevant range of penalties, on the identification of a core problem and on dynamic programming. The proposed approach turns out to be very effective in solving hard instances of PKP and compares favorably both to commercial solver CPLEX 12.5 applied to the ILP formulation of the problem and to the best available exact algorithm in the literature. Then we present a general inapproximability result and investigate several relevant special cases which permit fully polynomial time approximation schemes (FPTASs).

We consider the 0-1 Penalized Knapsack Problem (PKP). Each item has a profit, a weight and a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the items included in a solution. We propose an exact approach relying on a procedure which narrows the relevant range of penalties, on the identification of a core problem and on dynamic programming. The proposed approach turns out to be very effective in solving hard instances of PKP and compares favorably both to commercial solver CPLEX 12.5 applied to the ILP formulation of the problem and to the best available exact algorithm in the literature. Then we present a general inapproximability result and investigate several relevant special cases which permit fully polynomial time approximation schemes (FPTASs).

We consider the 0-1 Penalized Knapsack Problem (PKP). Each item has a profit, a weight and a penalty and the goal is to maximize the sum of the profits minus the greatest penalty value of the items included in a solution. We propose an exact approach relying on a procedure which narrows the relevant range of penalties, on the identification of a core problem and on dynamic programming. The proposed approach turns out to be very effective in solving hard instances of PKP and compares favorably both to commercial solver CPLEX 12.5 applied to the ILP formulation of the problem and to the best available exact algorithm in the literature. Then we present a general inapproximability result and investigate several relevant special cases which permit fully polynomial time approximation schemes (FPTASs).

In this thesis we study several computer implementations of the thinning algorithm, a new method for generating non-homogeneous Poisson processes. The method is valid for Poisson processes with any given intensity function. The basic thinning algorithm is modified to exploit several refinements which reduce computer execution time by approximately one-third. The basic and modified thinning programs are compared with the Poisson decomposition and gap-statistics algorithm, which is easily implemented for Poisson processes with intensity functions of the form exp(a sub 0 + a sub 1t + a sub 2 t-squared. The thinning programs are competitive in both execution time and computer memory requirements. One program implementation generates the events in a Poisson process one at a time; another program implements the algorithmic refinements which improve efficiency.

The design of algorithm plays an important role in operations research in general and dynamic programming in particular. In most dynamic programming algorithms, formalism of computing, data structure and complexity analysis does not appear. One of the objectives of this research is to provide such a formalism. Converging branch, diverging branch, feed-forward loop and feedback loop systems are considered. In each case, first, a high level algorithm followed by the detailed computer algorithm is described. Formulas for storage and computational complexities for each computer algorithm are derived. Finally algorithms are implemented on VAX-11/780 computers using UCSD PASCAL. (Author)

ADA066345

ADA066345

The Next Generation (NextGen) transport aircraft configurations being investigated as part of the NASA Aeronautics Subsonic Fixed Wing Project have more control surfaces, or control effectors, than existing transport aircraft configurations. Conventional flight control is achieved through two symmetric elevators, two antisymmetric ailerons, and a rudder. The five effectors, reduced to three command variables, produce moments along the three main axes of the aircraft and enable the pilot to control the attitude and flight path of the aircraft. The NextGen aircraft will have additional redundant control effectors to control the three moments, creating a situation where the aircraft is over-actuated and where a simple relationship does not exist anymore between the required effector deflections and the desired moments. NextGen flight controllers will incorporate control allocation algorithms to determine the optimal effector commands and attain the desired moments, taking into account the effector limits. Approaches to solving the problem using linear programming and quadratic programming algorithms have been proposed and tested. It is of great interest to understand their relative advantages and disadvantages and how design parameters may affect their properties. In this paper, we investigate the sensitivity of the effector commands with respect to the desired moments and show on some examples that the solutions provided using the l2 norm of quadratic programming are less sensitive than those using the l1 norm of linear programming.

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

The investigators associated with the research reported herein have been supported by ARO almost continuously over the past 20 years, from the inception of their early development of penalty function and branch and bound methodology. This report covers results developed from 1 January 1974 to 31 May 1981 under two ARO contracts and one grant, results dealing mainly with nonconvex programming, second order algorithms and sensitivity and stability methodology. Technical details are omitted, the main intent being the provision of a concise chronicle of the major accomplishments.

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.

Topics addressed include: data mapping and rotational algorithms for the Massively Parallel Processor (MPP); Parallel Pascal language; documentation for the Parallel Pascal Development system; and a description of the Parallel Pascal language used on the MPP.

Although large amounts of programming knowledge are available to human programmers in the form of books and articles, very little of this knowledge is available in a form suitable for use by a machine in performing programming tasks automatically. The principal goal of the research reported here is the explication of programming knowledge to a sufficient level of detail that it can be used effectively by a machine. The programming task considered in this experiment is that of constructing concrete implementations of abstract algorithms in the domain of symbolic programming. Knowledge about several aspects of symbolic programming has been expressed as a collection of four hundred refinement rules. The rules deal primarily with collections and mappings and ways of manipulating such structures, including several enumeration, sorting and searching techniques. The principal representation techniques covered include the representation of sets as linked lists and arrays (both ordered and unordered), and the representation of mappings as tables, sets of pairs, property list markings, and inverted mappings (indexed by range element). In addition to these general constructs, many low-level programming details are covered (such as the use of variables to store values).

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

We propose Chunks and Tasks, a parallel programming model built on abstractions for both data and work. The application programmer specifies how data and work can be split into smaller pieces, chunks and tasks, respectively. The Chunks and Tasks library maps the chunks and tasks to physical resources. In this way we seek to combine user friendliness with high performance. An application programmer can express a parallel algorithm using a few simple building blocks, defining data and work objects and their relationships. No explicit communication calls are needed; the distribution of both work and data is handled by the Chunks and Tasks library. This makes efficient implementation of complex applications that require dynamic distribution of work and data easier. At the same time, Chunks and Tasks imposes restrictions on data access and task dependencies that facilitates the development of high performance parallel back ends. We discuss the fundamental abstractions underlying the programming model, as well as performance and fault resilience considerations. We also present a pilot C++ library implementation for clusters of multicore machines and demonstrate its performance for sparse blocked matrix-matrix multiplication.

This paper investigates session programming and typing of benchmark examples to compare productivity, safety and performance with other communications programming languages. Parallel algorithms are used to examine the above aspects due to their extensive use of message passing for interaction, and their increasing prominence in algorithmic research with the rising availability of hardware resources such as multicore machines and clusters. We contribute new benchmark results for SJ, an extension of Java for type-safe, binary session programming, against MPJ Express, a Java messaging system based on the MPI standard. In conclusion, we observe that (1) despite rich libraries and functionality, MPI remains a low-level API, and can suffer from commonly perceived disadvantages of explicit message passing such as deadlocks and unexpected message types, and (2) the benefits of high-level session abstraction, which has significant impact on program structure to improve readability and reliability, and session type-safety can greatly facilitate the task of communications programming whilst retaining competitive performance.

This paper investigates session programming and typing of benchmark examples to compare productivity, safety and performance with other communications programming languages. Parallel algorithms are used to examine the above aspects due to their extensive use of message passing for interaction, and their increasing prominence in algorithmic research with the rising availability of hardware resources such as multicore machines and clusters. We contribute new benchmark results for SJ, an extension of Java for type-safe, binary session programming, against MPJ Express, a Java messaging system based on the MPI standard. In conclusion, we observe that (1) despite rich libraries and functionality, MPI remains a low-level API, and can suffer from commonly perceived disadvantages of explicit message passing such as deadlocks and unexpected message types, and (2) the benefits of high-level session abstraction, which has significant impact on program structure to improve readability and reliability, and session type-safety can greatly facilitate the task of communications programming whilst retaining competitive performance.

For paving the way to novel applications in quantum simulation, computation, and technology, increasingly large quantum systems have to be steered with high precision. It is a typical task amenable to numerical optimal control to turn the time course of pulses, i.e. piecewise constant control amplitudes, iteratively into an optimised shape. Here, we present the first comparative study of optimal control algorithms for a wide range of finite-dimensional applications. We focus on the most commonly used algorithms: GRAPE methods which update all controls concurrently, and KROTOV-type methods which do so sequentially. Guidelines for their use are given and open research questions are pointed out. --- Moreover we introduce a novel unifying algorithmic framework, DYNAMO (dynamic optimisation platform) designed to provide the quantum-technology community with a convenient MATLAB-based toolset for optimal control. In addition, it gives researchers in optimal-control techniques a framework for benchmarking and comparing new proposed algorithms to the state-of-the-art. It allows for a mix-and-match approach with various types of gradients, update and step-size methods as well as subspace choices. Open-source code including examples is made available at http://qlib.info.

This paper studies the problem of finding a (1+eps)-approximation to positive semidefinite programs. These are semidefinite programs in which all matrices in the constraints and objective are positive semidefinite and all scalars are non-negative. Previous work (Jain and Yao, FOCS'11) gave an NC algorithm that requires at least Omega((1/eps)^13\log^{13}n) iterations. The algorithm performs at least Omega(n^\omega) work per iteration, where n is the dimension of the matrices involved, since each iteration involves computing spectral decomposition. We present a simpler NC parallel algorithm that requires O((1/eps)^4 \log^4 n \log(1/eps)) iterations. Moreover, if the positive SDP is provided in a factorized form, the total work of our algorithm can be bounded by \otilde(n+M), where M is the total number of nonzero entries in the factorization. Our algorithm is a generalization of Young's algorithm and analysis techniques for positive linear programs (Young, FOCS'01) to the semidefinite programming setting.

Donald Knuth is one of the greatest and most impactful computer scientists and mathematicians ever. He is the recipient in 1974 of the Turing Award, considered the Nobel Prize of computing. He is the author of the multi-volume work, the magnum opus, The Art of Computer Programming. He made several key contributions to the rigorous analysis of the computational complexity of algorithms. He popularized asymptotic notation, that we all affectionately know as the big-O notation. He also created the TeX typesetting which most computer scientists, physicists, mathematicians, and scientists and engineers use to write technical papers and make them look beautiful. This conversation is part of the Artificial Intelligence podcast. This episode is presented by Cash App. Download it & use code "LexPodcast": Cash App (App Store): https://apple.co/2sPrUHe Cash App (Google Play): https://bit.ly/2MlvP5w INFO: Podcast website: https://lexfridman.com/ai Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/category/ai/feed/ Full episodes playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Clips playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 EPISODE LINKS: The Art of Computer Programming (book): https://amzn.to/39kxRwB OUTLINE: 0:00 - Introduction 3:45 - IBM 650 7:51 - Geeks 12:29 - Alan Turing 14:26 - My life is a convex combination of english and mathematics 24:00 - Japanese arrow puzzle example 25:42 - Neural networks and machine learning 27:59 - The Art of Computer Programming 36:49 - Combinatorics 39:16 - Writing process 42:10 - Are some days harder than others? 48:36 - What's the "Art" in the Art of Computer Programming 50:21 - Binary (boolean) decision diagram 55:06 - Big-O notation 58:02 - P=NP 1:10:05 - Artificial intelligence 1:13:26 - Ant colonies and human cognition 1:17:11 - God and the Bible 1:24:28 - Reflection on life 1:28:25 - Facing mortality 1:33:40 - TeX and beautiful typography 1:39:23 - How much of the world do we understand? 1:44:17 - Question for God CONNECT: - Subscribe to this YouTube channel - Twitter: https://twitter.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - Medium: https://medium.com/@lexfridman - Support on Patreon: https://www.patreon.com/lexfridman Source: https://www.youtube.com/watch?v=2BdBfsXbST8 Uploader: Lex Fridman

Donald Knuth is one of the greatest and most impactful computer scientists and mathematicians ever. He is the recipient in 1974 of the Turing Award, considered the Nobel Prize of computing. He is the author of the multi-volume work, the magnum opus, The Art of Computer Programming. He made several key contributions to the rigorous analysis of the computational complexity of algorithms. He popularized asymptotic notation, that we all affectionately know as the big-O notation. He also created the TeX typesetting which most computer scientists, physicists, mathematicians, and scientists and engineers use to write technical papers and make them look beautiful. This conversation is part of the Artificial Intelligence podcast. This episode is presented by Cash App. Download it & use code "LexPodcast": Cash App (App Store): https://apple.co/2sPrUHe Cash App (Google Play): https://bit.ly/2MlvP5w INFO: Podcast website: https://lexfridman.com/ai Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/category/ai/feed/ Full episodes playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Clips playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 EPISODE LINKS: The Art of Computer Programming (book): https://amzn.to/39kxRwB OUTLINE: 0:00 - Introduction 3:45 - IBM 650 7:51 - Geeks 12:29 - Alan Turing 14:26 - My life is a convex combination of english and mathematics 24:00 - Japanese arrow puzzle example 25:42 - Neural networks and machine learning 27:59 - The Art of Computer Programming 36:49 - Combinatorics 39:16 - Writing process 42:10 - Are some days harder than others? 48:36 - What's the "Art" in the Art of Computer Programming 50:21 - Binary (boolean) decision diagram 55:06 - Big-O notation 58:02 - P=NP 1:10:05 - Artificial intelligence 1:13:26 - Ant colonies and human cognition 1:17:11 - God and the Bible 1:24:28 - Reflection on life 1:28:25 - Facing mortality 1:33:40 - TeX and beautiful typography 1:39:23 - How much of the world do we understand? 1:44:17 - Question for God CONNECT: - Subscribe to this YouTube channel - Twitter: https://twitter.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - Medium: https://medium.com/@lexfridman - Support on Patreon: https://www.patreon.com/lexfridman Source: https://www.youtube.com/watch?v=2BdBfsXbST8 Uploader: Lex Fridman

Donald Knuth is one of the greatest and most impactful computer scientists and mathematicians ever. He is the recipient in 1974 of the Turing Award, considered the Nobel Prize of computing. He is the author of the multi-volume work, the magnum opus, The Art of Computer Programming. He made several key contributions to the rigorous analysis of the computational complexity of algorithms. He popularized asymptotic notation, that we all affectionately know as the big-O notation. He also created the TeX typesetting which most computer scientists, physicists, mathematicians, and scientists and engineers use to write technical papers and make them look beautiful. This conversation is part of the Artificial Intelligence podcast. This episode is presented by Cash App. Download it & use code "LexPodcast": Cash App (App Store): https://apple.co/2sPrUHe Cash App (Google Play): https://bit.ly/2MlvP5w INFO: Podcast website: https://lexfridman.com/ai Apple Podcasts: https://apple.co/2lwqZIr Spotify: https://spoti.fi/2nEwCF8 RSS: https://lexfridman.com/category/ai/feed/ Full episodes playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 Clips playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOeciFP3CBCIEElOJeitOr41 EPISODE LINKS: The Art of Computer Programming (book): https://amzn.to/39kxRwB OUTLINE: 0:00 - Introduction 3:45 - IBM 650 7:51 - Geeks 12:29 - Alan Turing 14:26 - My life is a convex combination of english and mathematics 24:00 - Japanese arrow puzzle example 25:42 - Neural networks and machine learning 27:59 - The Art of Computer Programming 36:49 - Combinatorics 39:16 - Writing process 42:10 - Are some days harder than others? 48:36 - What's the "Art" in the Art of Computer Programming 50:21 - Binary (boolean) decision diagram 55:06 - Big-O notation 58:02 - P=NP 1:10:05 - Artificial intelligence 1:13:26 - Ant colonies and human cognition 1:17:11 - God and the Bible 1:24:28 - Reflection on life 1:28:25 - Facing mortality 1:33:40 - TeX and beautiful typography 1:39:23 - How much of the world do we understand? 1:44:17 - Question for God CONNECT: - Subscribe to this YouTube channel - Twitter: https://twitter.com/lexfridman - LinkedIn: https://www.linkedin.com/in/lexfridman - Facebook: https://www.facebook.com/lexfridman - Instagram: https://www.instagram.com/lexfridman - Medium: https://medium.com/@lexfridman - Support on Patreon: https://www.patreon.com/lexfridman Source: https://www.youtube.com/watch?v=2BdBfsXbST8 Uploader: Lex Fridman

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

The class $(r,2)$-CSP, or simply Max 2-CSP, consists of constraint satisfaction problems with at most two $r$-valued variables per clause. For instances with $n$ variables and $m$ binary clauses, we present an $O(n r^{5+19m/100})$-time algorithm which is the fastest polynomial-space algorithm for many problems in the class, including Max Cut. The method also proves a treewidth bound $\tw(G) \leq (13/75+o(1))m$, which gives a faster Max 2-CSP algorithm that uses exponential space: running in time $\Ostar{2^{(13/75+o(1))m}}$, this is fastest for most problems in Max 2-CSP. Parametrizing in terms of $n$ rather than $m$, for graphs of average degree $d$ we show a simple algorithm running time $\Ostar{2^{(1-\frac{2}{d+1})n}}$, the fastest polynomial-space algorithm known. In combination with ``Polynomial CSPs'' introduced in a companion paper, these algorithms also allow (with an additional polynomial-factor overhead in space and time) counting and sampling, and the solution of problems like Max Bisection that escape the usual CSP framework. Linear programming is key to the design as well as the analysis of the algorithms.

Journal of Research of the National Bureau of Standards

Learn Data Structures and Algorithms with Golang is a practical guide for developers looking to deepen their understanding of core computer science concepts using the Go programming language. Through hands-on examples and clear explanations, this book helps you implement efficient algorithms and master essential data structures to write faster, more scalable code in Go.