1,720,969 research outputs found
A computational analysis of the optimal power flow problem
No full text1 online resource (PDF, 20 pages, includes illustrations)Alzalg, Baha; Anghel, Catalina; Gan, Wenying; Huang, Qing; Rahman, Mustazee; Shum, Alex. (2012). A computational analysis of the optimal power flow problem. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/181205
Semidefinite Programming and Combinatorial Optimization
No full textIn modern convex optimization, the class of optimization that is an immediate enlargement of second-order cone optimization is semidefinite optimization. Semidefinite programming (SDP for short) problems, which include linear programming problems and second-order cone programming problems as special cases, are a class of convex optimization problems in which the variable is not a vector that is required to be nonnegative, but rather a symmetric matrix that is required to be positive semidefinite. This book chapter is devoted to studying SDP problems
Recurrences
No full textIn this book chapter, we present some techniques for solving recurrences. There are five efficient methods for solving recurrences: the guess-and-confirm method, the iteration method, the recursion-tree method, the generating functions method, and the master method. In this chapter, we present the first four methods. The fifth method (master method) is beyond the scope of the book
Mathematical Logic
No full textThe precise definition of “logic” is quite broad and literally, hundreds of logics have been studied by philosophers, mathematicians, and computer scientists. When most people say "logic", they mean either propositional logic or predicate logic. The propositional logic is the classical one, in which there are two possible truth values (i.e., true and false). The predicate logic extends propositional logic with the ability to speak explicitly about objects and their properties. In this book chapter, we first study propositional logic, the simplest and most abstract logic we can learn. After that, we look at the predicate logic
Optimization Over Symmetric Cones Under Uncertainty
No full textThesis (Ph.D.), Department of Mathematics, Washington State UniversityWe introduce and study two-stage stochastic symmetric programs (SSPs) with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a logarithmic barrier decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proceeds by showing that the log barrier associated with the recourse function of SSPs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao's results [Math. Program., Ser. A, 90:507-536, 2001] for two-stage stochastic linear programs, and Mehrotra and Ozevin's results [SIAM J. of Optimization, 18(1): 206-222, 2007] for two-stage stochastic semidefinite programs (SSDPs). We also present another class of polynomial-time decomposition algorithms for SSPs based on the volumetric barrier. While this extends the work of Ariyawansa and Zhu [Mathematics of computation, 80: 1639-1661, 2011] for SSDPs, our analysis is based on utilizing the advantage of the special algebraic structure associated with the symmetric cone not utilized in [Mathematics of computation, 80: 1639-1661, 2011]. As a consequence, we are able to significantly simplify the proofs of central results. We then describe four applications leading to the SSP problem where, in particular, the underlying symmetric cones are second-order cones and rotated quadratic cones.Department of Mathematics, Washington State Universit
Set-Theoretic Structures
No full textMuch of mathematical discrete structures are written in terms of sets. In this book chapter, we introduce basic concepts of set theory. After introducing mathematical induction as a method for proving mathematical statements, we study set-theoretic structures such as sets, relations, partitions, and functions
Second-Order Cone Programming
No full textIn modern convex optimization, the class of optimization that is an immediate enlargement of linear programming is not quadratic programming, but rather the so-called second-order cone programming. Second-order cone programming (SOCP for short) problems, which include linear programming problems and quadratic programming problems as special cases, are a class of convex optimization problems in which the variable is not a vector whose each of its components is required to be nonnegative, but rather a block vector whose each of its subvectors is required to reside in a second-order cone. So in an SOCP problem, we optimize a linear function over the intersection of an affine linear manifold with the Cartesian product of second-order cones. This chapter is devoted to studying SOCP problems
Array and Numeric Algorithms
No full textIn this book chapter, we present and analyze some standard array and numeric algorithms, such as array multiplication algorithms, array searching algorithms, array sorting algorithms, and Newton’s method algorithm for solving linear and nonlinear systems. We also describe and analyze the integer Euclidean algorithm (or Euclid’s algorithm), which is one of the oldest and simplest number theoretic algorithms
Counting
No full textIn this book chapter, we present some important counting principles, permutations, and combinations. The first section in this chapter introduces binomial coefficients and identities which will be used throughout the chapter
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