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Topic: Local optima

	Maxima and minima - Wikipedia, the free encyclopedia
		In order to be able to define local maxima and local minima, the function needs to take real values, and the concept of neighborhood must be defined on the domain of the function.
		One refers to a local maximum/minimum as to a local extremum (or local optimum), and to a global maximum/minimum as to a global extremum (or global optimum).
		For twice-differentiable functions in one variable, a simple technique for finding local maxima and minima is to look for stationary points, which are points where the first derivative is zero.
	en.wikipedia.org /wiki/Maxima_and_minima (607 words)

	Publications - NHDOJ
		Optima appears to have developed a corporate culture, led by management and acquiesced in by its trustees, which assumes that the delivery of health care is best left exclusively to the sole judgment of management.
		Optima referred to CMC and Elliot as "Optima East Campus" and "Optima West Campus."(5) Optima claims that its treatment of CMC and Elliot as a single combined hospital is justified because it could not achieve the cost efficiencies and quality improvements promised at the time of the merger without consolidating the two hospitals into one.
		Optima's decisions to reorganize the governance of the hospitals and to consolidate their acute care services at a single site are properly characterized as a decision to establish a single acute care facility as a successor to CMC and Elliot.
	www.state.nh.us /nhdoj/publications/optima1.html (14472 words)

	How many local optima? (Site not responding. Last check: 2007-10-12)
		I want to find out how many local optima a problem tends to have.
		I have code (in C) that could be used to find a local optimum and I have data for several test cases.
		The problem is to figure out how to use the above to estimate the number of local optima.
	www.cc.gatech.edu /classes/cs7100_97_fall/projects/tovey01.html (132 words)

	2.3 Local Search in Combinatorial Optimization
		Local search is based on the concept of a neighborhood.
		Local search can be seen as a walk on a directed graph whose vertices are solutions points and whose arcs connect neighboring points.
		In particular, if the graph is disconnected, then the neighborhood is not exact since there exist feasible points that would lead to local optima but not the global optima.
	www.cs.cmu.edu /afs/cs.cmu.edu/project/jair/pub/volume15/ambite01a-html/node9.html (786 words)

	Starter (Site not responding. Last check: 2007-10-12)
		OPTIMA batteries deliver a higher level of power to the starter in the critical first 10 seconds of the vehicle starting cycle.They feature a patented SPIRALCELL Technology allowing for the lowest internal resistance of any battery on the market today.
		This decreased internal resistance,along with the completely sealed case,is the key to Optima's unparalleled high power delivery, increased cycle life and fast recharge capability for fast recovery in short driving cycles.
		OPTIMA's unique SPIRALCELL design offers the greatest technology in providing a strong and clean power source that far surpasses any of today's filled lead/acid batteries.
	www.optimabatteries.com /publish/optima/americas0/en/config/product_info/automotive/starter.html (336 words)

	Multivariate optimization: local optima (Site not responding. Last check: 2007-10-12)
		We may be able to tell whether a stationary point is a local maximum, a local minimum, or neither by examining the second derivative of the function at the stationary point.
		In this case, information about the signs of the higher order derivatives may tell us whether a point is a local maximum or a local minimum.
		As for a function of a single variable, a stationary point of a function of many variables may be a local maximizer, a local minimizer, or neither, and we may be able to distinguish the cases by examining the second-order derivatives of the function at the stationary point.
	www.chass.utoronto.ca /~osborne/MathTutorial/LON.HTM (1144 words)

	Exercises on local optima
		Is x = 1 a local maximum or a local minimum of the function f (x) =
		Find all the local maxima and minima and all the global maxima and minima, if any, of the following functions.
		For each local maximum that you find, determine, if possible, whether it is a global maximum.
	www.economics.utoronto.ca /osborne/MathTutorial/LONX1.HTM (300 words)

	Why We Need Lots of Optimization Algorithms
		Because of noise and other factors, it is often the case in practice that the simple global structure of a problem is masked by parasitic local optima.
		We do not regard these local optima as representing important features of the model and so, if we could somehow smooth the objective function, we would be more confident that we had determined the extremum we seek.
		Sometimes the local optima represent significant features in the model (for example, fundamental ambiguities can exist in inverse calculations; the global optimum may not be fundamentally more significant than other local optima).
	coool.mines.edu /report/node3.html (810 words)

	2002 Kia Optima Prices and Review (Site not responding. Last check: 2007-10-12)
		Optima is based on the Hyundai Sonata produced by Kia's South Korean corporate parent.
		Optima uses Sonata's 4-cyl and V6 engines and comes in LX and SE models.
		We can connect you with a local Kia dealer that not only meets the highest standards of customer service, but one who will provide you with a great new-car price quote on a 2002 Kia Optima -- hassle free.
	auto.consumerguide.com /Auto/New/reviews/full/index.cfm/id/20857/... (730 words)

	Examples (Site not responding. Last check: 2007-10-12)
		In the other case there were 10 local optima distributed at random in the graph and a point was in the domain of attraction of a given local optimum if and only if it was closer to that optimum than to any other, and once again searching was really easy.
		From this spread we can infer that the landscape has only a few local optima and that they are close together in the space.
		Work is underway to estimate landscape characteristics of simulated instances of the phylogeny problem for various values of parameters, such as the overall amount of evolutionary change on the tree, transformations of the data to yield smoother landscapes, number of taxa, effects of random noise in the data etc.
	taxonomy.zoology.gla.ac.uk /~mac/landscape/examples.html (476 words)

	HGSP: Hopfield Networks
		It is possible to keep track of a particular set of points (such as the local optima) while highlighting other things.
		If you have previously saved the local optima as a marked group, you can see which can be reached.
		If the local optimum has neighbours (in the HSGP topology) that are coloured yellow, then these points form the cycle.
	www.itee.uq.edu.au /~janetw/hsgp/hsgp-hopfield.html?print=1 (550 words)

	[No title]
		In general, these methods are restricted to finding local optima for problems that satisfy the underlying assumptions and can only be used for global optimization in special cases.
		Further, function surfaces in large-scale complex problems often have large number of dimensions and can be very rugged, trapping local-search methods in local optima and often requiring restarts to bring the search to a new region.
		The search trajectory is based on two counteracting forces: local gradient information that drives the search to a local optimum, and a deterministic trace that leads the search out of the local optimum once it gets there.
	manip.crhc.uiuc.edu /Wah/rsnovel.html (695 words)

	ICASE Colloquium: Steven Cox on July 14, 2000 (Site not responding. Last check: 2007-10-12)
		The conceptual design of aircraft often entails a large number of nonlinear constraints that result in a nonconvex feasible design space and multiple local optima.
		SQP is a local optimizer, while Snymans algorithm is capable of moving through shallow local minima.
		The DIRECT algorithm is found to be the most cost effective for locating the global optimum of functions with true local optima while DOT was efficient at moving through noise.
	www.icase.edu /colloq-v3/data/colloq.Cox.Steven.2000.7.14.html (199 words)

	Global Versus Local Optima
		The smallest local minimum of an objective function is called the global minimum, and the largest local maximum of an objective function is called the global maximum.
		One way to find out whether the objective function has more than one local optimum is to run various optimizations with a pattern of different starting points.
		A local maximizer has the largest value in a feasible neighborhood and a negative definite Hessian.
	www.ualberta.ca /CNS/RESEARCH/Software/SAS.old/iml/chap11/sect4.htm (392 words)

	[No title]
		First-Order (Necessary) Condition Looking back at the previous diagram, it’s apparent that local optimum points represent stationary points in the function, where the slope of the function changes from positive to negative (for a maximum), or negative to positive (for a minimum).
		It’s clear therefore that we need second-order conditions to determine whether a point at which the first derivative is zero is a local maximum, a local minimum, or a point of inflection.
		Thus we conclude that at a local minimum point the second derivative is positive.
	www.lmu.ac.uk /lbs/epia/people/beachill/quants/quantch4.doc (1692 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		As N increases, the fitness of local optima falls toward the mean fitness of the entire space.
		As the genotype veers away from local optima, the drop is likely to be more precipitous as K increases.
		Conversely, on a rugged landscape, optima tend to be surrounded by deep chasms, thus selection forces are strong and types tend to not fall off peaks due to mutation.
	www.societyforchaostheory.org /tutorials/00004/NonlinearDynamics101-NK.doc (2380 words)

	Multivariate optimization: local optima
		A stationary point may be a local maximizer, a local minimizer, or neither.
		As for functions of a single variable, we may be able to tell whether a stationary point is a local maximum or a local minimum by examining second-order derivatives of the function at the stationary point.
		If x* is a local minimizer then H(x*) is positive semidefinite.
	free.prohosting.com /cepr/data/adveco/lon.html (778 words)

	[No title]
		The key issue that drives the effectiveness of local search algorithms is the size and design of the neighborhood function.
		Therefore, to overcome the trappings of local optima, smart methods of escaping from local optima must be obtained.
		One possible approach is a cyclical simulated annealing algorithm that cycles through a set of temperature parameters, and hence, allows for an easier and more intelligent mechanism to escape local optima.
	www.engr.wisc.edu /ie/colloquia/Jacobsoncolloquium.htm (342 words)

	EE372 Projects -- Project Suggestions (Site not responding. Last check: 2007-10-12)
		The idea is that different compression algorithms or distortion measures may be more appropriate for different local image types and that better overall compression can be obtained by identifying the type and using the suitable algorithm.
		Try to evaluate the claim by using known locally optimal algorithms with multiple starting points or by using random jumps from local optima as in simulated annealing, or by using sufficiently simple sources and small dimensions to allow exhaustive location of all local minima.
		Can you find any properties of the true optima or even good local optima that would assist a clustering algorithm in avoiding poor local optima, i.e., additional optimality properties for the Lloyd algorithm past the usual nearest neighbor, centroid, and zero-probability boundary rules.
	www.stanford.edu /class/ee372/projects_1.html (2336 words)

	Measuring the Landscape (Site not responding. Last check: 2007-10-12)
		We can estimate, for each instance of the phylogeny problem, the number and distribution of local optima, the size of the domain of attraction of each local optimum we find, the distribution of values of the optimality criterion (c.f.
		In the "ideal" case, we have just one optimum which is therefore globally optimal, or a few local optima which are very close to each other in the space.
		When there are many local optima, spread far apart in the space, that means that in terms of the tree adjacency rules (tree perturbations) we have chosen, the optimal trees are not similar.
	taxonomy.zoology.gla.ac.uk /~mac/landscape/measuring.html (777 words)

	AR1 command changes 8/98
		This is in spite of such multiple optima being common when lagged dependent variables are present.
		This allows optima with rho > 1 to be located, since iterations rarely can pass rho=1 (which is usually a singular point where the objective function is discontinuous).
		Quick and reliable iterations are needed to accurately locate local and thus global optima.
	www.stanford.edu /~clint/bench/ar1_898.htm (966 words)

	Non-oblivious local search
		Now, a different type of local search can be obtained by using a different objective function to direct the search, i.e., to select the best neighbor at each iteration.
		Local optima of the standard objective function f are not necessarily local optima of the different objective function.
		Theorems 7-8 of [22] state that the performance ratio for any oblivious local search algorithm with a d-neighborhood for MAX-2-SAT is 2/3 for any d = o(n), while non-oblivious local search with an 1-neighborhood achieves a performance ratio 3/4.
	www.jea.acm.org /TURING/Vol2Nbr2/node8.html (440 words)

	[No title]
		C ABSTRACT: C Simulated annealing is a global optimization method that distinguishes C between different local optima.
		You'll see (i) it never C escapes from the local optima (in annealing terminology, it C quenches) & (ii) the step length (VM) will be quite small.
		There are two optima: C F(.864,1.23) = 16.0817 (the global minumum) and F(2.35,-.319) = 20.9805.
	www.rpi.edu /AFS/home/03/hajela/public/simann.f (3961 words)

	generation5 - Optimizing Functions on the Real Numbers (Site not responding. Last check: 2007-10-12)
		The function may be discontinuous, non-differentiable, have a fractal structure, change over time as we are trying to optimize it and/or only give us an expected value instead of its actual value each time we evaluate it.
		In the search for global optima, a great problem is getting stuck in local optima.
		As with gradient descent, if a population is showing little improvment because it is stuck in a local optima or flat region of the fitness function, and/or its population lacks diversity, adding more random noise to the parents or including parents with lower fitness may be helpful.
	www.generation5.org /content/2004/OptimizingRealFunctions.asp?Print=1 (4620 words)

	IZA - Discussion Papers
		In practice, however, a succession of marginal changes to a limited number of policy instruments are implemented, until no further improvement is feasible.
		I show that there is history dependence, in that several local optima may be reached, and which one is reached depends on initial conditions.
		Finally, for a given economy and a given tax code, I define the latter’s average performance as the average value of the social welfare function across all the local optima.
	www.iza.org /de/webcontent/publications/papers/viewAbstract?dp_id=480 (239 words)

	local optimum (Site not responding. Last check: 2007-10-12)
		Note: Some search methods may get trapped in a local optimum and miss the global optimum.
		Go to the Dictionary of Algorithms and Data Structures home page.
		Paul E. Black, "local optimum", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/localoptimum.html (97 words)

	Exercises on local optima
		Draw a figure showing the regions of the (x,y) plane at which this function has positive values and the regions at which it has negative values.
		Is the point (x,y) = (0,0) a local minimum of f ?
		Find all the local maxima and minima of the following functions.
	free.prohosting.com /cepr/data/adveco/lonx1.html (142 words)

	CEPR Discussion Paper Abstracts
		I consider the outcome of such a tatonment process for a government that wants to optimally set taxes given a tax code with a fixed number of brackets.
		History dependence is stronger (ie there are more local optima), the more complex the design of economic policy, ie the greater the number of tax brackets.
		It is also typically stronger, the greater the interaction of policy instruments with one another — which in my model is equivalent to agents having a more elastic labour supply behaviour.
	www.cepr.org /pubs/new-dps/dplist.asp?dpno=3339 (335 words)

	sam roweis : nldr
		It has no local optima and the size of the system it must solve scales with the number of local models rather than the number of original data points, making it more efficient than model-free algorithms such as Isomap or LLE.
		Our local linear models are represented by a mixture of factor analyzers, and the ``global coordination'' of these models is achieved by adding a regularizing term to their objective function.
		LLE uses local symmetries and linear reconstructions to compute low dimensional, neighborhood preserving embeddings of multivariate data.
	www.cs.toronto.edu /~roweis/nldr.html (915 words)

	[No title]
		First, a proper local search scheme must be chosen.
		Because random variables are used in the algorithm, the outcome of different trials may vary even for the exact same choice of cooling schedule.
		Moreover, the convergence to the global optima of simulated annealing is only achieved when algorithm proceeds to infinite number of iterations.
	www.cs.wisc.edu /~dyer/cs540/notes/Simulated_Annealing.ppt (514 words)

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