Exp AstronDOI 10.1007/s10686-014-9400-7

ORIGINAL ARTICLE

Optimal strategies for surveys with WSO-UV/ISSISA mathematical programming and heuristic approaches

J. Yanez ·Ana I. Gomez de Castro ·Javier Lopez-Santiago

Received: 9 November 2013 / Accepted: 4 June 2014© Springer Science+Business Media Dordrecht 2014

Abstract Astronomical surveys provide an amazing amount of information for theunderstanding of specific astrophysical problems, from the large scale structure ofgalaxy clusters, to the formation of stars. Large scale surveys are often comple-mented with smaller surveys based on many pointings in areas of special interest.The World Space Observatory-Ultraviolet (WSO-UV) is the coming 2-m classfacility for UV astronomy. The imaging instrument in WSO-UV is ISSIS (Imagingand Slitless Spectroscopy Instrument). A fraction of the ISSIS science will involvemapping and analyzing areas of the sky extending many arc minutes to observeclusters of galaxies, stellar associations, star-forming regions (either galactic orextragalactic) and other interesting astronomical objects. ISSIS field of view is small(< 2 arcmin) and survey strategies need to be defined, well in advance, to optimizethe scientific output. We present the analysis done by our team to define optimalstrategies for mapping large areas of the sky where relevant sources may be scatteredin a non-random manner. The optimization problem is addressed with differentmathematical programming models: Continuous, Binary and Sample-Binary Mathe-matical Programming Models, as well as through heuristic approaches. A set ofoptimization algorithms is defined for various surveying strategies based on mathe-matical approaches.

Keywords Astronomical instrumentation · Methods and techniques ·Methods: miscellaneous

J. Yanez · A. I. Gomez de Castro (�) · J. Lopez-SantiagoFacultad de Matematicas, Universidad Complutense de Madrid,Plaza de las Ciencias 3, 28040 Madrid, Spaine-mail: aig@mat.ucm.es

J. Yanez · A. I. Gomez de Castro · J. Lopez-SantiagoAEGORA Research Group, Universidad Complutense de Madrid,Plaza de las Ciencias 3, 28040 Madrid, Spain

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1 Introduction

During the past 15 years there has been a substantial effort to collect a large databaseof panchromatic information of the sky. Infrared surveys starting from the IRASmission [12], to the most recent missions like SPITZER [2] or survey projects like2MASS[18]. X-ray surveys starting with EINSTEIN [5] and following with ROSATand the coming mission Spectrum-X. Optical surveys and redshift surveys likeSLOAN DR9 [1] and the last of the surveys, the UV survey of an 80 % of the sky bythe mission GALEX [10]. These surveys provide an amazing amount of information,often difficult to ingest, that has to be processed for the understanding of specificastrophysical problems, from the origin of the large scale structure, to the galacticstructure, the formation of stars and the chemical evolution of the Universe. They arealso used to seek for candidates of especially relevant objects such as brown dwarfs,black holes, neutron stars, binary systems and AGNs.

WSO-UV is the coming mission for UV astronomy [15–17]. After the completionof the Hubble Space Telescope (HST) mission, WSO-UV will be the only 2-mclass facility that will permit to run a proper analysis of the UV sky uncoveredby the GALEX mission [9]. WSO-UV is a Russian led mission with a significantSpanish contribution in terms of instrumentation and ground segment definition andoperation. The mission is part of the Russian Space program for 2005-2015 and itwill be launched before the end of the decade. The telescope will be orbiting ingeosynchronous orbit in a near circular orbit and will be operated as an internationalscientific observatory. There will be two operations centers (at Moscow and Madrid)that will act as science centers and provide access to the Archive to the world-widescientific community. The scientific objectives of the WSO-UV emission are tostudy: 1) the chemical evolution of the nearby Universe (z < 2), 2) the physicsof astronomical engines, 3) the magnetic activity of stars and, in general, stellarphysics, 4) the late stages of the pre-main sequence evolution of Solar-type starsand the impact of the stellar UV radiation in the young planetary disk evolutionand exoplanets atmospheres and 5) astrochemical processes in heavily UV irradiatedenvironments.

WSO-UV instrumentation is designed for targeted observations. The WSO-UVSpectrographs (WUVS) include:

- The far-UV high-resolution echelle spectrograph (VUVES) operating in the1150-1750 A range, with resolution R∼55,000.

- The near-UV high-resolution echelle spectrograph (UVES), 1750-3100 A range,resolution R∼55,000.

- The Long Slit Spectrograph (LSS) for low-resolution (R∼1000), long slitspectroscopy. Both spatial resolution and the width of the slit will be 0.5 arcsec.

They are equipped with CCD detectors that provide high sensitivity for far UVspectroscopy.

The imaging instrument of WSO-UV is ISSIS (Imaging and Slitless SpectroscopyInstrument), the first UV imager to be flown to high Earth orbit, above the Earthgeocorona [6–8]. Hence, the UV background will be dominated by the zodiacalcontribution and the diffuse galactic background due to dust-scattered starlight [11].

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The instrument has been designed to make full benefit of the heritage left by theGALactic Evolution eXplorer (GALEX) mission. GALEX spatial resolution is ∼4.2arcsec and has very moderate spectroscopic capabilities. ISSIS spatial resolution willbe ≤ 0.1 arcsec. The Fine Guiding System of the WSO-UV telescope will guaranteea very high pointing stability (better than 0.1 arcsec at 3σ ). Moreover, ISSIS willbe equipped with gratings for slitless spectroscopy with resolution 500, in the full1150 A- 3200 A spectral range. In imaging mode, ISSIS effective area is about 10times that of the GALEX imagers.

ISSIS is designed to be an instrument for analysis of weak UV point sources orclumpy extended sources. UV imaging instruments have been often equipped withprisms or very low dispersion grisms. The rapid decay of the resolution of prismssuch as the available in the Solar Blind Channel (SBC) of the Advanced CameraSystem (ACS) makes very difficult its use to map extended line emission at wave-lengths above some 1350 A. As the transmittance of narrow band filters in the farUV is ≤3 % , integral-field low-resolution spectroscopy is left as the main tool tomap nebular emission. ISSIS gratings will make feasible to use the powerful UVdiagnostic tools to determine the location of dusty blobs and measure electron den-sities and temperatures. However, ISSIS field of view is very limited (< 2 arcmin)since it makes use of Micro Channel Plate (MCP) detectors. MCP detectors consistof three basic components: 1) a photocathod sensitive to the wavelength range ofinterest which is deposited in a transparent window, 2) an MCP that amplifies theelectronic signal by a factor or 105 to 106 depending on the voltage and 3) a read-outelectronics. MCP detectors have several disadvantages, as lifetime and low quantumefficiency compared with CCDs. However, they are solar blind1 which makes of themthe detector of choice for the observation of weak UV sources with strong opticalemission. MCP building principles set a maximum size that in the case of ISSISshows as a small field of view if high resolution is required, as it is.

A fraction of the ISSIS science will involve mapping and analyzing areas of thesky extending many arcminutes to observe clusters of galaxies, stellar associationsor star-forming regions (either galactic or extragalactic). Previous knowledge of theareas to be observed will be available from the X-ray, optical and infrared surveys.For areas well above/below the galactic plane, this information can be retrieveddirectly from the GALEX survey. This provides an ”a priori” insight on the distribu-tion of sources and opens the possibility to define optimal observation strategies thatguarantee that ISSIS science time is used efficiently.

In this article, we present the analysis done by our team to define optimal strategiesfor the mapping of large areas of the sky where relevant sources may be scatteredin non-random manner. The description of the mathematical problem is provided inSection 2, and the heuristic approaches in Section 3. The analysis of the efficiency

1Solar blind detectors are characterized for having a negligible sensitivity to photons with wavelengthsabove some 3200 A . This is a common requirement for detectors intended to observe the UV radiationfrom solar-like stars since the photospheric flux is very high compared with the UV radiation produced bythe high atmospheric layers.

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achieved by this optimization process is shown in Section 4. The article concludeswith a brief summary in Section 5. The mathematical programming models areavailable upon request as Matlab procedures.

2 The mathematical programming models

The mathematical problem can be stated as follows: given a large area of the sky,let O be the set of n astronomical objects of interests; any of such objects oi , wherei ∈ {1, 2, . . . , n}, is characterized by its International Celestial Reference System(ICRS) coordinates: the right ascension αi and the declination δi .

Let m be the number of telescope pointings to be made. Any of such pointings j ∈{1, 2, . . . , m} has two parameters: the ICRS coordinates α

pj and δ

pj . An additional

parameter is the angular size, the radius of the field of view of the instrument ρ,which is the same for all pointings. The spherical distance between any two points(αi, δi) and (αj , δj ) is given by:

sph d(P, P ′) = acos(sin(δj ) sin(δi) + cos(δj ) cos(δi) cos(αj − αi))

The mathematical problem consists in selecting the m points{(α

pj , δ

pj ), j ∈

{1, 2, . . . , m}} that maximize the number of astronomical objects in the total field.For example, let us assume that there are 9 sources to be observed distributed as

shown in Fig. 1. Only 6 objects can be observed with the two pointings depicted inthe figure: (α

p

1 , δp

1 ) = (1.5, 1.0) and (αp

2 , δp

2 ) = (3.5, 2.5).

Fig. 1 Illustrative example. One of the possible solutions

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In general, this problem can be stated through different mathematical program-ming models, as it is shown below.

2.1 The continuous mathematical programming model

The natural decision variable could be the ICRS coordinates of the m telescopepointings. With this definition of the decision variable -denoted as {(αp

j , δpj )

j = 1, 2, . . . , m}- any object oi ∈ O will be observed if there exists one of thesepointings j such that the spherical distance between Pj = (α

pj , δ

pj ) and oi = (αi, δi)

is less than ρ.Given n astronomical objects, with ICRS coordinates {(αi, δi), i = 1, . . . , n} and

the decision variable

Pj =(α

pj , δ

pj

)∀j ∈ {1, . . . , m}

and introducing the auxiliary variables:

- zi ∈ {0, 1}, for 1 ≤ i ≤ n, which states whether the object oi ∈ O is observed(zi = 1) or not (zi = 0).

- γi ∈ R, for 1 ≤ i ≤ n, is the minimum spherical distance from object oi ∈ O tosome center Pj = (α

pj , δ

pj ), with 1 ≤ j ≤ m.

- tij ∈ {0, 1}, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, are technical binary variables thatassure γi attains the minimum value of spherical distances.

the mathematical programming model is the following:

Max∑n

i=1 zi

Subject to :(1) γi ≤ sph d(Pj , oi) ∀i ∈ {1, . . . , n}

∀j ∈ {1, . . . , m}(2) γi ≥ ∑m

j=1 tij sph d(Pj , oi) ∀i ∈ {1, . . . , n}(3)

∑mj=1 tij = 1 ∀i ∈ {1, . . . , n}

(4) γi ≤ ρ + (1 − zi)π ∀i ∈ {1, . . . , n}

The objective function computes the number of observed objects. Constraints (1)guaranty that γi is lower than the spherical distances between oi and every pointing;constraints (2) and (3) sets γi as the maximum of all the available values, i.e., theminimum of the spherical distances; otherwise, the trivial values γi = 0 ∀i will alsobe valid; finally, constraints (4) allows zi = 1 provided that the spherical distance islower than ρ; otherwise, the constraint is relaxed and the variable zi is set to 0.

This model has 2 · m + n continuous variables, n · (m + 1) binary variables and(m + 3) · n constraints. Note that the constraints associated with the function sph d

are non-linear. As a result, this is a very hard computational problem and will notbe considered in the following. Let us define Z

c, the optimum objective function for

this model.

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2.2 The binary mathematical programming model

The computational program can be simplified if the m ICRS coordinates of thepointings are forced to coincide with ICRS coordinates of some targets. With thisapproach, the decision variable is defined as

x(i) ={

1 if telescope points towards oi = (αi, δi)

0 otherwise∀i ∈ {1, .., m}

The optimal solution of this model will not be better than the previous one(see Section 2.1) since the decision variable is restricted to a finite number of values.However, this approach could give solutions close to the optimal for large n.

Given this decision variable, let

zi ∈ {0, 1}, ∀i ∈ {1, 2, . . . n}be the previous auxiliar variable which states whether the object oi ∈ O is observed(zi = 1) or not (zi = 0).

Then, the mathematical (binary) programming model is the following:

Max∑ n

i=1 zi

Subject to z(i) ≤ ∑ nj=1 neig(i, j)x(j) ∀i ∈ {1, 2, . . . n}

The neighborhood matrix can be computed in advance and allows the linearity ofthe mathematical model. Each element of the matrix, neig(i, j), is set equal to 1 ifthe spherical distance between any pair of objects: oi and oj is lower than ρ and it isset equal to 0 otherwise:

neig(i, j) ={

1 if sph d(oi, oj ) ≤ ρ

0 otherwise∀i, j ∈ {1, 2, . . . n}

This model is linear, but it has 2 · n binary variables and n linear constraints. Let

us define Zb, the optimum objective function for this model.

It is worth noticing that this binary programming problem is a NP -hard problem[3]. Henceforth, to solve it in a bounded computing time requires that the size of thedecision variable is made as small as feasible; with this idea, the following model isintroduced.

2.3 The sample-binary mathematical programming model

The computing time will be reduced significantly if selection of the possible point-ings is done randomly from the original list of targets to be observed as shown inFig. 4. In this manner, the sample size will be nr with (m ≤ nr ≤ n); moreover, the

suboptimal objective function defined in this way will be closer to Zb

as nr → n.Let nr be the number of astronomical objects which can be pointed at by the

telescope; in this way, the decision variable is

x(i) ={

1 if telescope points towards oi = (αi, δi)

0 otherwise∀i ∈ {i1, . . . inr }

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where

{i1, . . . inr } ⊂ {1, 2, . . . n}The mathematical (sample-binary) programming model is the following:

Max∑ n

i=1zi

Subject to z(i) ≤ ∑ nr

j=1neig(i, ij )x(j) ∀i ∈ {1, 2, . . . n}

This model has n+ nr binary variables and n linear constraints. Let us define Zsb

nr

as the optimum objective function for this model.In all cases, it is verified that

Zc ≥ Z

b ≥ Zsb

nr

3 Heuristic approaches

The solution of the last mathematical programming model (the sample-binary) canbe considered as heuristic, in the sense that seeks good (i.e. near-optimal) solutions ata reasonable computational cost without being able to guarantee optimality [4, 13].Other heuristic approaches can be applied to the problem as it will be shown below.

3.1 A greedy algorithm

The simplest heuristic procedure to construct a good solution in an optimal way step-by-step is the Greedy algorithm. If the quality of this solution is acceptable for thescientific requirements, the search procedure would stop here; otherwise, the solutionwould be used as input (initial solution) for other heuristics.

Given the problem data,(n, {(αi, δi), i = 1, . . . , n}, m, ρ

)

the auxiliary matrix neig is introduced:

neig(i, j) ={

1 if sph d(oi, oj ) ≤ ρ

0 otherwise∀i, j ∈ {1, 2, . . . n}

A greedy solution xg is constructed including at each step a new pointing to theastronomical object with the greatest number of non-observed astronomical objectslocated at a spherical distance lower than ρ. Note that for a given object oi , the objectsoj located at a spherical distance lower than ρ are identified by neig(i, j) = 1.

In order to avoid overlapping, the number of non previously observed astronomicalobjects which can be observed from any astronomical object oi is introduced as:

obs∗(i) ≡n∑

j=1

((1 − z(j))neig(i, j)

),∀i ∈ {1, 2, . . . , n}

(the observed targets j (z(j) = 1) do not increase the variable obs∗(i))

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Thus, the greedy algorithm is:•xg(i) = 0 ∀i = 1, . . . , n There are no pointings•z(j) = 0 ∀j = 1, . . . , n Objects are not observable•obs∗(i) ≡ ∑n

j=1 neig(i, j) ∀i ∈ {1, 2, . . . , n}• for k = 1, m The best suitable pointing is selected◦ Let h be such that obs∗(h) = maxi | xg(i)=0{obs∗(i)}◦ xg(h) = 1◦ z(j) = min{z(j) + neig(h, j); 1} z(j) are updated◦ obs∗(i) = ∑n

j=1(1 − z(j))neig(i, j) obs∗(i) are updatedend

For the example depicted in Fig. 1, this greedy algorithm will give the solutiondepicted in the Fig. 2. The number of observed targets is equal to 7.

3.2 Other heuristics approaches

In general, it is feasible to improve a non-optimal solution using a neighborhoodalgorithm. For instance, let X be the set of feasible solutions for a combinatorialproblem, for any solution x ∈ X, a set N(x) ⊂ X of neighbors can be defined,called the neighborhood of x. The basic neighborhood algorithm starts with an initialsolution x0 (will be valid a greedy one, for instance); then, the best neighbor, denotedas x1 is selected:

z(

x1)= max

x∈N(x0)

{z(x)

}

and this process is repeated until z(xk+1) = z(xk).For the illustrative example depicted in Fig. 1, a neighborhood algorithm will give

the solution depicted in Fig. 3. The number of observed targets is equal to 8.

Fig. 2 Illustrative example. A greedy algorithm solution

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Fig. 3 Illustrative example. A local neighborhood algorithm solution

The greedy and local neighborhood algorithms described above do not guaranteethat the solution is optimal; they are a constructive and an improvement heuristics,respectively.

Alternative approximated solutions are the modern heuristics or metaheuristics. Ametaheuristic is a high-level problem-independent algorithmic framework that pro-vides a set of guidelines or strategies to develop heuristic optimization algorithms.

Fig. 4 Case study. The data

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Table 1 The sample-binaryprogramming model applied tothe example of Section 4.1

nr CPU Time Znr nr CPU Time Znr

(sec) (sec)

300 86.21 2328 500 90.29 2380

300 1105.57 2316 500 90.29 2380

300 95.31 2325 500 90.14 2390

300 83.91 2316 500 1011.17 2384

300 84.49 2342 500 93.28 2350

Following Rothlauf, see [14]: In contrast to standard improvement heuristics whichusually only perform improvement steps, modern heuristics also allow inferior solu-tions to be generated during search. Therefore, we want to define modern heuristicsas improvement heuristics that 1) can be applied to a wide range of different problems(they are general-purpose methods), and 2) use during search both intensification(exploitation) and diversification (exploration) phases.

The goal of intensification steps is to improve the quality of solutions. In contrast,diversification explores new areas of the search space, thus accepting also completeor partial solutions that are inferior to the currently obtained solution(s).

Some well known metaheuristics are Simulated annealing, Tabu Search, GeneticAlgorithms, etc. [13].

4 Computational experiences: a case of study

To study the impact of the heuristics, metaheuristics and optimal solutions describedabove in the optimization of sky surveying strategies, we have selected an area of thegalactic plane, including the young galactic cluster IC 2391. The region is centeredin the cluster and has a radius of 2 deg. This region has been chosen to reflect a typ-ical case of Astronomical interest: the observation of a stellar cluster in a crowdedstellar field. IC 2391 is a nearby young stellar cluster that extends > 1 square degreesin the sky. The density of stars in the cluster is not homogeneous. The best choice toobserve the maximum number of members (and possible members) of the cluster isto use a wide-field camera as, e.g. the Auxiliary-port Camera (ACAM) of the WilliamHerschel Telescope (WHT) at the Observatorio del Roque de Los Muchachos

Table 2 The sample-binaryprogramming model applied tothe example of Section 4.1

nr CPU Time Znr nr CPU Time Znr

(sec) (sec)

750 93.62 2391 1000 30448.39 *

750 8817.16 2385 1000 96.81 (#) 2441

750 11196.86 * 1000 97.37 2442

750 106.44 2405 1000 110.03 (#) 2444

750 95.97 2406 1000 106.70 2432

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(La Palma, Spain) or multi-fiber spectrographs as FLAMES mounted on the VLT ofthe Paranal Observatory in Chile. The field of view of such instruments is typically10′ × 10′, although smaller telescopes may reach 20′ × 20′. The observation of theentire region is time consuming and the best strategy to observe as much candidatesas possible must be investigated.

A 3-degrees radius region centered on the core of Ic 2391 has n = 13168objects. Their Right Ascension (in radians) are in the interval [2.1979, 2.3426]; theirDeclinations (also in radians) are in the interval [−0.9673,−0.8802]. They aredepicted in Fig. 4.

Two possible observational strategies have been considered:

- Observing with an instrument with a large field of view but only allowing for fewpointings, as it would be the case of using ACAM or FLAMES (Section 4.1).

- Observing with ISSIS which has a smaller field of view but allows for severalpointings (Section 4.2).

As will be shown below, the mathematical treatment to best identify the optimalsolution is different.

4.1 Case A: large field of view and few pointings

For this case, we have assumed m = 25 and ρ = π/1080 that corresponds to anangular radius of 10 arcmin. We have attempted to find the optimal solution with thethree main algorithms described in Section 3.

Fig. 5 Optimal solution (Zg = 2530) for ρ = 10 minutes and m = 25

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Fig. 6 Optimal solution (Zb = 483) for ρ = 2 minutes and m = 50

Among mathematical programming algorithms, the Binary Mathematical Modelsshows to be unable to converge to a solution in 12 hours of CPU time using Matlab2-based program in a PC Intel Core i5 CPU 2.50 GHz. However, the Sample-BinaryMathematical Programming Model is able to find a good solution in roughly 2400 sas shown in Tables 1 and 2.

It can be noticed that in the objective function Znr , the number of observedastronomical objects, increases with increasing nr . However, if nr is made grow arbi-trarily, the computing time increases to the point that, in some cases, the procedure isinterrupted with a warning message maximum time exceeded; these cases are markedwith an asterisk (∗) in Table 2. Occasionally, Znr may not be integer due to someinternal Matlab numerics. Matlab permits to define an integer tolerance parameterfor this purpose and the results are marked with a # symbol in Table 2.

There is an upper limit to the applicability of this scheme. For nr ≥ 1500 thisalgorithm is unable to find a feasible solution in a reasonable computing time.

2Matlab is a programming language developed by MathWorks, see www.mathworks.com

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In these cases a greedy-algorithm is the most straightforwards procedure to finda good solution. When applied to our test case, an objective function z = ∑n

i=1z(i) = 2530 is obtained. This solution is plotted in Fig. 5; the solution is achieved injust 12.4 second of CPU-time.

4.2 Case B: small field of view and many pointings

For this case, we assume m = 50 and ρ = π/5400, that corresponds to an angularradius of 2 arcmin.

The Binary Mathematical Programming Model introduced in Section 2.2 can beused to compute the optimal solution in a reasonable time. The solution is plotted inFig. 6; the objective function is

∑z(i) = 483, and the CPU time of 292.48 sec.

As in Case A, also a greedy algorithm can be used to approximate the best solu-tion. In this case we recover the best solution, Zg = 483, but this is not generallyguaranteed. The CPU-time is, however, slightly larger when using a the greedy algo-rithm (306.02 seconds) than when applying the Binary Programming model. Thisslight difference is related to the program coding and though it is not significant inthis case, it may be important for other problems.

5 Conclusions

Mathematical programming models provide a true optimal solution and they can beimplemented to define the general survey strategy for ISSIS, given its small fieldof view. For larger fields of view (∼ 10′), forbiddenly long computing times arerequired to identify the optimal solution and heuristic approaches are better suitedfor optimization strategies. For the two cases studied in this work, the neighborhoodlocal algorithms do not improve the greedy solutions. Simulated annealing schemesand genetic algorithms have also been tested and have not improved the results whilethe computing performance is significantly worse in terms of CPU time.

The main result of this work is that there is not a unique mathematical treatmetto find the best solution for astronomical surveys optimization. A broad battery ofalgorithms need to be considered and taylored for the case of study. For small fieldof views, binary models provide an exact solution in a reasonable computer time.However, as the number of objects in the field of view grows, as the field of viewbecomes wider, heuristic approaches are needed. In the cases analyzed in this work,a greedy algorithm is found to be powerful enough to provide a good solution. Forharder problems, the use of more sophysticated technics (metaheuristics) may berequired.

This problem can be easily generalized to take the priority of a given astronomicalobservation into account just by assigning weights to the targets in the optimizationprocedure. In such a case, the new objective function

∑zi can be transformed

(without a significative increase in computing time) into a function of the type∑wizi for a given family of known weights {wi , i = 1, . . . , n}.

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Acknowledgements This research was partially supported by grant AYA2011-29754-C03-01 from theGovernment of Spain.

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