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Topic: Quadratic programming

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	Ch7. Lecture Notes
		Be able to formulate a quadratic programming model of portfolio analysis developing and incorporating estimates of mean and variances of returns among investment alternatives and subsequent measures of the expected value and variance of total returns from alternative portfolios.
		Quadratic programming is a common appraoch to represent nonlinearities in optimization problems.
		The quadratic programming model minimizes the variance of returns to the entire portfolio (composed of different proportions of stock 1 and stock 2) subject to meeting a specified level of expected returns from the portfolio.
	www.ndsu.nodak.edu /instruct/swandal/AGEC339f/ch22121.htm (1063 words)

	Optimization Problem Types - Linear and Quadratic Programming
		A linear programming (LP) problem is one in which the objective and all of the constraints are linear functions of the decision variables.
		A quadratic programming (QP) problem has an objective which is a quadratic function of the decision variables, and constraints which are all linear functions of the variables.
		A quadratic with a semi-definite Hessian is still convex: It has a bowl shape with a "trough" where many points have the same objective value.
	www.solver.com /probtype2.htm (781 words)

	[No title]
		Mixed-integer Programming: Mixed-integer programmings problems are linear pro- gramming problems where some of the variables are constrained to be inte- gers.
		For example, a mixed-integer programming problem may be concerned with the number of esats on an airplane or airplanes on a route.
		Quadratic Programming: Quadratic programming problems solved by OSL have a convex quadratic objective function and linear constraints.
	www.uic.edu /depts/adn/infwww/txt/v2729012.txt (750 words)

	Quadratic programming - Wikipedia, the free encyclopedia
		A quadratic programming problem has at least one of the following kinds of constraints:
		Quadratic programming with one negative eigenvalue is NP-hard, Panos M. Pardalos and Stephen A. Vavasis in Journal of Global Optimization, Volume 1, Number 1, 1991, pg.15-22.
		AIMMS Optimization Modeling AIMMS — include quadratic programming in industry solutions (free trial license available)
	en.wikipedia.org /wiki/Quadratic_programming (257 words)

	Publications (in mathematics) list for Henry Wolkowicz
		Semidefinite programming (SDP) relaxations for the quadratic assignment problem (QAP) are derived using the dual of the (homogenized) Lagrangian dual of appropriate equivalent representations of QAP.
		Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables.
		This problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem.
	orion.math.uwaterloo.ca /~hwolkowi/henry/reports/ABSTRACTS.html (11057 words)

	Nonlinear Programming FAQ (Site not responding. Last check: 2007-10-08)
		The word "Programming" is used here in the sense of "planning"; the necessary relationship to computer programming was incidental to the choice of name.
		Hence the phrase "NLP program" to refer to a piece of software is not a redundancy, although I tend to use the term "code" instead of "program" to avoid the possible ambiguity.
		Semidefinite Programming is a generalization of linear programming to the space of block diagonal, symmetric, positive semidefinite matrices.
	www.faqs.org /faqs/nonlinear-programming-faq (4771 words)

	Nonlinear Programming FAQ
		Communicating with a nonlinear programming code can be particularly tedious and error-prone, especially if you have to write programs in a language like Fortran or C to compute function (and maybe gradient) values for your objective and constraints.
		Schittkowski, by Hock and Schittkowski, and by Torn and Zilinskas.
		A generator for quadratic programming test problems is described by Calamai, Vicente and Judice in "A new technique for generating quadratic programming test problems," Mathematical Programming 61 (1993) 215-231.
	www-unix.mcs.anl.gov /otc/Guide/faq/nonlinear-programming-faq.html (5846 words)

	[No title]
		On quadratic and $O(\sqrt{n}L)$ convergence of a predictor- corrector algorithm for LCP (Y. Ye and K. Anstreicher), Mathematical Programming 62, (1993) 537-552.
		On the finite convergence of interior-point algorithms for linear programming, Mathematical Programming 57, (1992) 325-335.
		On the quadratic convergence of the $O(\sqrt{n}L)$-iteration homogeneous and self-dual linear programming algorithm (F. Wu, S. Wu, and Y. Ye), manuscript, Academia Sinica and the University of Iowa (1992).
	dollar.biz.uiowa.edu /col/ye/paper.html (2383 words)

	Newton-KKT Interior-Point Methods for Indefinite Quadratic Programming (Site not responding. Last check: 2007-10-08)
		Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems.
		Our algorithms are adapted from previously proposed algorithms for convex quadratic programming and general nonlinear programming.
		Global and local quadratic convergence are proved under nondegeneracy assumptions for both algorithms.
	www.montefiore.ulg.ac.be /~absil/Publi/indefQP.htm (158 words)

	SQ^P, Sequential Quadratic Constrained Quadratic Programming (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		we develop a local quadratic model at a current approximate minimizer in conjunction with a trust region.
		Asymptotically, finding the local minimizer of the quadratic model is equivalent to applying Newton's method to the stationarity condition.
		500 Linear and Nonlinear Programming (context) - LUENBERGER - 1984
	citeseer.ifi.unizh.ch /kruk98sqp.html (627 words)

	Linear Programming
		While the basic simplex algorithm is not too difficult to program, there is a considerable art to producing an efficient implementation capable of solving large linear programs.
		A linear program is called an integer program when all its variables have integrality constraints, or a mixed integer progam if some of them do.
		Pascal implementations of the revised and dual simplex methods for linear programming, as well as cutting plane and explicit enumeration algorithms for integer programming, are provided in [SDK83].
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK3/NODE141.HTM (1514 words)

	quadratic programming (Site not responding. Last check: 2007-10-08)
		QP solves the standard quadratic programming problem: min{1/2x'Qx - x'R}, subject to constraints: Ax = B and Cx >= D, with bounds: Xl <= x <= Xu, where x is a vector of unknown coefficients, and Q, R, A, B, C, D, Xl, and Xu are known matrices.
		Constrained least squares is a special case of the the quadratic programming problem.
		The Mean-Variance and Mean-Semivariance models are quadratic programming problems where Q is the covariance matrix of a portfolio of stocks, bonds, options, etc., and R is a vector of their mean values.
	www.scientific-solutions.ch /tech/gauss/qp.html (340 words)

	Maximizing Quadratic Programs: Extending Grothendieck's Inequality (Site not responding. Last check: 2007-10-08)
		This quadratic programming problem can be seen as an extension to that of maximizing x^TAy (where y's components are also ±1).
		The study of this type of quadratic program arose from a desire to approximate the maximum correlation in correlation clustering.
		We can also guarantee that our quadratic programming algorithm returns a solution to the MAXCUT problem that has a significant advantage over a random assignment.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/focs/2004/2228/00/2228toc.xml&DOI=10.1109/FOCS.2004.39 (232 words)

	Methods - Operations Research Models and Methods (Site not responding. Last check: 2007-10-08)
		Separable programming is important because it allows a convex nonlinear program to be approximated with arbitrary accuracy with a linear programming model.
		A linearly constrained optimization problem with a quadratic objective function is called a quadratic program (QP).
		Because of its many applications, quadratic programming is often viewed as a discipline in and of itself.
	www.me.utexas.edu /~jensen/ORMM/supplements/methods/nlpmethod/nlp_intro.html (151 words)

	AMCA: An efficient algorithm for nonlinear optimization based on sequential quadratic programming by M. Chuedoung (Site not responding. Last check: 2007-10-08)
		Sequential quadratic programming generally has superlinear convergence and is considered to be one of the best methods for solving nonlinear constrained programming problems.
		However, in the traditional sequential quadratic programming method, a large matrix, the Hessian matrix, needs to be calculated and stored in each iteration, which limits its application to large-scale optimization problems.
		In this paper, we present an efficient sequential quadratic programming algorithm in which, the quadratic programming subproblem can be implemented in such a way that the Hessian matrix is not needed explicitly.
	at.yorku.ca /c/a/d/r/14.htm (164 words)

	Strategies for Solving Mathematical Programming Problems
		In linear programming, both the objective function and the constraints are linear.
		Pure network programming problems are linear programming problems in which the columns of the constraint matrix have exactly two nonzero elements, one equal to 1 and the other equal to -1.
		A mixed-integer programming problem is a mathematical programming problem in which some of the variables are restricted to integer values.
	www.mang.canterbury.ac.nz /people/shane/oslweb/features/featur07.htm (9732 words)

	Sequential Quadratic Programming
		The sequential quadratic programming (sequential QP) algorithm is a generalization of Newton's method for unconstrained optimization in that it finds a step away from the current point by minimizing a quadratic model of the problem.
		The strategy based on (1.3) makes the decision about which of the inequality constraints appear to be active at the solution internally during the solution of the quadratic program.
		A somewhat different algorithm is obtained by making this decision prior to formulating the quadratic program.
	www-fp.mcs.anl.gov /otc/Guide/OptWeb/continuous/constrained/nonlinearcon/section2_1_1.html (576 words)

	Solver Technology - Linear Programming and Quadratic Programming
		The Large-Scale SQP Solver for the Premium Solver Platform uses a state-of-the-art implementation of an active set method for solving linear (and quadratic) programming problems, which fully exploits sparsity in the model to save time and memory, and uses modern matrix factorization methods for numerical stability.
		It is able to solve extremely large quadratic programming problems, if sufficient memory is available.
		However, it is appropriate only for positive definite quadratic objectives (when minimizing; negative definite when maximizing).
	www.solver.com /technology2.htm (866 words)

	Introduction (Site not responding. Last check: 2007-10-08)
		By allowing the position of the starting slice to be determined by the algorithm, we are allowing for the possibility of slicing that might coincide with part features, as demonstrated in Figure 3.
		By approximating the objective function as a quadratic function at each iteration, the algorithm intelligently chooses values for the variables that would put it at the minimum for that quadratic.
		The implementation was based on the CFSQP algorithm (C code for Feasible Sequential Quadratic Programming) developed at the University of Maryland.
	www.me.cmu.edu /faculty1/shimada/gm98/project/matt/project/report.html (1486 words)

	Sequential Quadratic Programming Method (Site not responding. Last check: 2007-10-08)
		The sequential quadratic programming method computes an approximate solution to a general constrained optimization problem by solving a sequence of quadratic programming problems obtained by applying Newton's method to find a critical point of the Lagrangian function.
		The steps of the sequential quadratic programming method are then carried out sequentially by repeatedly clicking on NEXT or on the currently highlighted step.
		This project is sponsored by the Computational Science and Engineering Program at the University of Illinois at Urbana-Champaign.
	www.cse.uiuc.edu /eot/modules/optimization/SQP_ConstrainedOptimization (250 words)

	WORMS Brian's Digest: Software - Quadratic Programming
		I'm looking for c/c++ - code to do quadratic programming (linear equality and inequality constraints), where the quadratic function to minimize is _not_ convex.
		Assume the problem proposed is an unconstrainted quadratic zero-one programming problem.
		The max cut problem is usually phrased in terms of a graph and modelled as a homogeneous quadratic over +-1 variables.
	www.worms.ms.unimelb.edu.au /digest/software/qp.html (1235 words)

	NEOS Guide - Quadratic Programming (Site not responding. Last check: 2007-10-08)
		MOSEK - linear programming and convex optimization (including convex quadratic programming).
		SQOPT - large-scale linear and convex quadratic programming.
		SNOPT - large-scale linear, quadratic, and nonlinear programming problems (including nonconvex quadratic programming.
	www-fp.mcs.anl.gov /otc/Guide/SoftwareGuide/Categories/quadprog.html (58 words)

	A decomposition method for quadratic programming
		We discuss the algorithms used in the Optimization Subroutine Library for the solution of convex quadratic programming problems.
		The basic simplex algorithm for convex quadratic programming is described.
		We then show how the simplex method for linear programming can be used in a decomposition crash procedure to obtain a good initial basic solution for the quadratic programming algorithm.
	domino.research.ibm.com /tchjr/journalindex.nsf/d9f0a910ab8b637485256bc80066a393/74eed842d057eb1885256bfa00685c77?OpenDocument (74 words)

	[No title]
		Information related to Semidefinite Programming is at ftp://orion.uwaterloo.ca/pub/henry/teaching/co769g/readme.html, which includes a pointer to some software.
		There is a code by Lieven Vandenberghe & Stephen Boyd at ftp://isl.stanford.edu/pub/boyd/semidef_prog for semidefinite programming which can be used to solve many nonlinear, convex optimization problems; includes full C source (which calls LAPACK), which can be used directly or via matlab mex file interfaces, matlab examples, and documentation.
		Available via anonymous FTP from rtfm.mit.edu in /pub/usenet/sci.answers/nonlinear-programming-faq There's a mail server on that machine, so if you don't have FTP privileges, you can send an e-mail message to mail-server@rtfm.mit.edu containing: send usenet/sci.answers/nonlinear-programming-faq as the body of the message to receive the latest version (it is posted on the first working day of each month).
	orion.math.uwaterloo.ca /~hwolkowi/henry/teaching/w99/367.w99/faq (3515 words)

	The Math Forum - Math Library - Linear Progrmng (Site not responding. Last check: 2007-10-08)
		Answers to questions such as: "What is Linear Programming?" "Where is there a good code to solve LP problems?" "Oh, and we also want to solve it as an integer program." "I wrote an optimization code.
		This is true for most linear programs, and it is becoming increasingly true for nonlinear and integer forms.
		Research into algorithms for linear programming which approach the solution through the interior of the feasible polygon, rather than moving around the boundary from vertex to vertex, as simplex methods do.
	mathforum.org /library/topics/linear_prog (1876 words)

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