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Angewandte Mathematik und Optimierung Schriftenreihe

Applied Mathematics and Optimization Series

AMOS # 59(2017)

Emily M. Craparo, Armin Fügenschuh

The Multistatic Sonar Location Problem and

Mixed-Integer Programming

Herausgegeben von der

Professur für Angewandte Mathematik

Professor Dr. rer. nat. Armin Fügenschuh

Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg

Fachbereich Maschinenbau

Holstenhofweg 85

D-22043 Hamburg

Telefon: +49 (0)40 6541 3540

Fax: +49 (0)40 6541 3672

e-mail: appliedmath@hsu-hh.de

URL: http://www.hsu-hh.de/am

Angewandte Mathematik und Optimierung Schriftenreihe (AMOS), ISSN-Print 2199-1928

Angewandte Mathematik und Optimierung Schriftenreihe (AMOS), ISSN-Internet 2199-1936

The Multistatic Sonar Location Problem and

Mixed-Integer Programming

Emily M. Craparo1and Armin F¨ugenschuh2

1Naval Postgraduate School, Operations Research Department, 1411 Cunningham

Road, Monterey, CA 93943, USA, emcrapar@nps.edu

2Helmut Schmidt University/University of the Federal Armed Forces Hamburg,

Holstenhofweg 85, 22043 Hamburg, fuegenschuh@hsu-hh.de

Abstract. A multistatic sonar system consists of one or more sources

that are able to emit underwater sound, and receivers that listen to the

direct sound as well as its reﬂected sound waves. From the diﬀerences

in the arrival times of these sounds, it is possible to determine the loca-

tion of surrounding objects. The propagation of underwater sound is a

complex issue that involves several factors, such as the density and pres-

sure of the water, the salinity and temperature level, as well as the pulse

length and volume and the reﬂection properties of the surface. These ef-

fects can be approximated by nonlinear equations. Furthermore, natural

obstacles in the water, such as the coastline, need to be taken into consid-

eration. Given a certain area of the ocean that should be endowed with

a sonar system for surveillance, we consider the task of determining how

many sources and receivers need to be deployed, and where they should

be located. We give an integer nonlinear formulation for this problem,

and several ways to derive an integer linear formulation from it. These

formulations are numerically compared using a test bed from coastlines

around the world and a state-of-the-art MIP solver (CPLEX).

Keywords: Integer Nonlinear Programming, Multistatic Sonar, Quadratic

Constraints, Linearization, Integer Linear Programming.

1 Introduction to Sonar

Sonar is a technique to detect objects that are under water or at the surface using

sound propagation. In active sonar systems, a sound is emitted from a source

and its echoes are detected by a receiver, revealing information about nearby

objects. Active sonar has been in use for nearly 100 years and has become a

key component of undersea detection. The basic operating principle of active

sonar is that acoustic energy is emitted from a source and its echoes are de-

tected by a receiver; these echoes reveal information about surrounding objects.

In a monostatic system, the source and the receiver are collocated in the same

place. Bistatic sonar uses a source and a receiver pair in diﬀerent locations. Mul-

tistatic sonar uses several sources and receivers simultaneously as a network. For

the surveillance of a large area of the ocean, a number of both types of devices

2 Craparo and F¨ugenschuh

must be deployed. This leads to an optimization problem to ﬁnd the least costly

multistatic network that is able to cover all of a desired area. No algorithm cur-

rently in the literature provides an optimal placement of an arbitrary number of

sources and receivers. In a discretized setting, we describe mathematical models

designed to determine the minimum-cost sensor layout that will cover a portion

of the ocean (a tile) by sonar surveillance, with adequate detection probability

throughout the tile. We model the physical properties of sound traveling between

sources, target, and receivers, the ocean (temperature, density, salinity) as well

as geometrical considerations (obstacles such as islands or coastlines). Details

are given in Section 2. We formulate an integer nonlinear program for the mul-

tistatic sonar source-receiver location problem and discuss several linearizations

in Section 3. We compare these formulations empirically using topological data

from coastal areas around the world and a state-of-the-art solver MIP solver and

give concluding remarks in Section 4.

2 Input Data

We obtain ocean topography data from [10]. At present, we do not use sea level

information and only distinguish in a binary fashion between the ocean (negative

elevation value) and the dry land (positive elevation value). A desired part of the

ocean and shoreline (a tile) is taken from the database. Since the resolution of the

data is too ﬁne to let each data pixel become a possible target/source/receiver

location, we aggregate the raw input data into larger rectangular areas (also

called grid cells). We then average the elevation data from all pixels within a cell

and apply the resulting elevation to the entire cell. Denote the set of rectangles

with negative elevation (i.e., those that are underwater) by G(for grid) and the

number of elements in Gby n:= |G|.

The sonar signal is characterized by the range of the day %0, which indicates

how quickly the signal diminishes as the target, source, and receiver become

farther apart. In a deﬁnite range (“cookie-cutter”) sensor model, a target in a

cell k∈Gis detected by a source placed in cell i∈Gand a receiver placed in

cell j∈Gwith probability pi,j,k ∈ {0,1}. Denote by di,j the Euclidean distance

between (the centers of) cell iand j. Necessary for detection (pi,j,k = 1) is that

the target kis inside the Cassini oval deﬁned by the equation di,k ·dk,j ≤%2

0,

c.f. [7]. If the target is too close to the line from source to receiver, then the

original signal and its reﬂection at the target become indistinguishable at the

receiver. This phenomenon is known as the direct blast eﬀect. The pulse length

κbdetermines the severity of this eﬀect, since longer pulses are more prone to

overlapping with the reﬂected signal. The direct blast zone is deﬁned by the

ellipsoid di,k +dk,j ≤di,j + 2κb, c.f. [6]. To account for the direct blast eﬀect, we

say that pi,j,k = 0, if the target lies within the direct blast zone. Additionally, if

an obstacle lies on either straight-line path of source to target, target to receiver,

or source to receiver, then pi,j,k = 0.

Multistatic Sonar Location Problem 3

The cost for each source is cs, and the cost for each receiver is cr. Typically,

cscr, i.e., a source is much more costly than a receiver, usually by a factor

of 5.

3 Model Formulations

All model formulations below have in common the binary decision variables

si, ri∈ {0,1}for each i∈G, which model the decision whether to place a source

(si= 1) or a receiver (ri= 1) in cell i. The objective (in all formulations) is to

minimize the total deployment cost, which we calculate as follows:

csX

i∈G

si+crX

j∈G

rj.(1)

An Integer Nonlinear Model. In the ﬁrst nonlinear formulation the binary

variables siand rjare multiplied in order to represent the joint decision of

placing a source at iand a receiver in j:

X

i∈GX

j∈G

pi,j,ksirj≥1,∀k∈G. (2)

Each constraint of (2) is a quadratic knapsack constraint. In general, for any

given k∈Gthe non-negative matrix (pi,j,k)i,j is indeﬁnite. The solver CPLEX

is able to process constraints of this form since version 12.6 [2]. Thus, the base run

for comparison with the other reformulation approaches is to solve the model:

min{(1)|(2); s, r ∈ {0,1}G}.(3)

The Oldest Linearization Technique. The ﬁrst documented linearization

of a product of binaries sirjby [3, 1] (and independently by others later on)

introduces a new binary variable hi,j ∈ {0,1}with hi,j = 1 if and only if si= 1

and rj= 1. In this method the constraints 2hi,j ≤si+rjand si+rj≤1 + hi,j

(for all i, j ∈G) are a linear description of this relationship. In our case, because

of the non-negativity of all pi,j,k, only the ﬁrst constraint is necessary. Thus the

ﬁrst linear version of (3) is

min (1),s.t. X

i∈GX

j∈G

pi,j,khi,j ≥1,∀k∈G, (4a)

2hi,j ≤si+rj,∀i, j ∈G, (4b)

s∈ {0,1}G, r ∈ {0,1}G, h ∈ {0,1}G×G.(4c)

Compared to the nonlinear integer formulation (3), this binary linear model has

an additional n2binary variables and n2constraints.

Standard Linearization of the Model. A linearization for sirjsimilar to

the previous one from [5] introduces continuous auxiliary variables hi,j ∈[0,1]

together with the constraints hi,j ≤si, hi,j ≤rjand si+rj≤1 + hi,j. This is

4 Craparo and F¨ugenschuh

perhaps the ﬁrst and most natural formulation to come to mind (and for good

reason: Padberg [9] showed that the constraints are facet deﬁning), and is hence

called “standard linearization.” As before, the third constraint is not required

in our case. Then, the second linear version of (3) is

min (1),s.t. X

i∈GX

j∈G

pi,j,khi,j ≥1,∀k∈G, (5a)

hi,j ≤si

hi,j ≤rj),∀i, j ∈G, (5b)

s∈ {0,1}G, r ∈ {0,1}G, h ∈[0,1]G×G.(5c)

Compared to the nonlinear binary formulation (3), this mixed-integer linear

model has an additional n2continuous variables and 2n2constraints.

Glover’s Linearization. To adapt a linearization technique from Glover [4],

we set Lj,k := Pi∈Gpi,j,k for all j, k ∈G, and the model reads:

min (1),s.t. X

j∈G

zj,k ≥1,∀k∈G, (6a)

Pi∈Gpi,j,ksi≥zj,k

Lj,krj≥zj,k ,),∀j, k ∈G, (6b)

s∈ {0,1}G, r ∈ {0,1}G, z ∈RG×G

+.(6c)

This model introduces n2additional continuous variables and 2n2additional

constraints (compared to (3)).

Oral-Kettani’s Linearization. Oral and Kettani [8] proposed two formula-

tions that come with n2additional continuous variables, but fewer constraints

compared to Glover’s formulation; namely, only n2(not counting the trivial

bound on zj,k as constraint). The ﬁrst of the two formulations is:

min (1),s.t. X

j∈G

(Lj,krj−zj,k )≥1,∀k∈G, (7a)

zj,k ≥Lj,krj−Pi∈Gpi,j,k si

Lj,k ≥zj,k,),∀j, k ∈G, (7b)

s∈ {0,1}G, r ∈ {0,1}G, z ∈RG×G

+.(7c)

The second Oral-Kettani linearization is:

min (1),s.t. X

j∈G X

i∈G

pi,j,ksi−zj,k !≥1,∀k∈G, (8a)

zj,k ≥Pi∈Gpi,j,ksi−Lj,k rj

Lj,k ≥zj,k,),∀j, k ∈G, (8b)

s∈ {0,1}G, r ∈ {0,1}G, z ∈RG×G

+.(8c)

Multistatic Sonar Location Problem 5

4 Computational Results and Conclusions

We compare the above six formulations on a test set of 22 instances. The ocean

topography data from various regions all over the world were extracted from

a global map, collected by Ryan et al. [10]. The computations were carried

out on a 2014 MacBookPro with 16 GB RAM and a 2.8 GHz Intel Core i7

processor. We set a time limit of 1,000 seconds and default settings of the solver

IBM ILOG CPLEX 12.7.1 otherwise. The results can be found in Table 1, with

the second Oral-Kettani formulation slightly ahead that of Glover, and CPLEX

failing to solve most instances within the time limit. An example result appears

in Figure 1.

Instance n(3) (4) (5) (6) (7) (8)

BabAlMandabStrait 29 1000.01 2.03 1.02 0.85 1.76 0.53

ChoctawhatcheeBay 31 1000.01 2.5 0.8 0.53 1.9 0.49

Dardanelles 19 3.04 0.24 0.07 0.1 0.07 0.04

EnglishChannel 48 1000.35 11.32 9.44 4.1 13.24 7.5

Falklandsund 57 1005.49 66.23 296.46 65.5 36.6 59.9

GulfOfAkaba 22 39.74 0.15 0.1 0.09 0.16 0.07

GulfOfFinland 37 1000.09 582.37 4.65 1.91 12.58 2.29

GulfOfSirte 45 1002.06 295.68 70.69 16.04 48.42 9.71

KarkinytskaGulf 34 1000.01 3.5 0.33 0.69 1.95 0.38

KerchStrait 36 1000.01 1.05 0.25 0.33 0.55 0.29

LagoDeMaracaibo 48 1000.02 4.45 1.46 1.44 2.35 1.6

Lesbos 30 1000.02 1.88 0.43 0.4 1.09 0.69

MontereyPeninsular 45 1000.26 12.19 5.06 4.29 12.74 2.66

NewYork 38 1000.52 6.59 1.25 1.57 8.27 2.83

OpenSea-Biscaya 54 1002.77 22.52 151.63 24.59 22.34 30.56

Oresund 71 1000.84 33.69 20.45 40.03 37.98 16.08

Ruegen 37 1000.02 34 7.3 2.94 45.32 1.19

Smalandsfarvandet 58 1000.73 229.88 31.74 26.09 32.02 7.53

Storebaelt 40 1000.25 57.58 10.63 2.26 12.66 2.57

StraitOfGibraltar 52 1000.49 28.66 43.62 7.72 70.45 15.63

StraitOfHormuz 41 1000.02 0.97 0.58 0.99 2.64 0.5

TaedongGang 39 1000.02 6.76 2.8 1.77 4.88 3.13

SUM 20056.77 1404.24 660.76 204.23 369.97 166.17

RANK 6 5 4 2 3 1

Table 1. Computational Results.

When facing a bilinear constraint of the type xTAy ≤bwith binary variable

vectors x, y and an indeﬁnite matrix A, several techniques for their linearization

were developed by researchers over the last ﬁve decades. Today, classical MILP

solvers (such as CPLEX) oﬀer features to automatically deal with such nonlinear

constraints, lifting the burden of going to the library from the user. As our results

demonstrate, it is still worthwhile to consider the knowledge of the past, and not

to blindly rely on the solver. Since it is unclear to determine a priori which of

6 Craparo and F¨ugenschuh

123456789

1

2

3

4

5

6

7

-60

4

2

23

2

1

1

1

1

1

2 1

1

1

1

1

1

1

1

2

2

11

2

2

1

3 3

2

3

2

1

1

1

4

2

2

1

12

1

2

1

22

Fig. 1. Left: The Monterey Peninsula area tile [10] as raw input data (365 cols, 285

rows). Right: Optimal placement of 2 sources (red circles) and 4 receivers (blue trian-

gles) on a 9x7 grid. Numbers ≥1 at each coordinate show multiplicity of coverage.

the method outperforms the others, it is necessary to implement and test all of

them.

References

1. E. Balas. Extension de l’algorithme additif `a la programmation en nombres en-

tiers et `a la programmation non lin´eaire. Technical report, Comptes rendus de

l’Acad´emie des Sciences, Paris, 1964.

2. C. Bliek, P. Bonami, and A. Lodi. Solving Mixed-Integer Quadratic Programming

problems with IBM-CPLEX: a progress report. In Proceedings of the Twenty-Sixth

RAMP Symposium Hosei University, Tokyo, October 16-17, 2014, 2014.

3. R. Fortet. L’alg`ebre de Boole et ses applications en recherche op´erationelle. Cahiers

du Centre d’´

Etudes de Recherche Op´erationelle, 4:5–36, 1959.

4. F. Glover. Improved Linear Integer Programming Formulations of Nonlinear Inte-

ger Problems. Management Science, 22(4):455–460, 1975.

5. F. Glover and E. Woolsey. Converting the 0-1 Polynomial Programming Problem

to a 0-1 Linear Program. Operations Research, 22(1):180–182, 1974.

6. M. Karatas and E.M. Craparo. Evaluating the Direct Blast Eﬀect in Multistatic

Sonar Networks Using Monte Carlo Simulation. In L. Yilmaz et al., editor, Pro-

ceedings of the 2015 Winter Simulation Conf. IEEE Press, Piscataway, NJ, 2015.

7. M. Karatas, E.M. Craparo, and A. Washburn. A Cost Eﬀectiveness Analysis of

Randomly Placed Multistatic Sonobuoy Fields. In C. Bruzzone et al., editor, The

International Workshop on Applied Modeling and Simulation, 2014.

8. M. Oral and O. Kettani. A Linearization Procedure for Quadratic and Cubic

Mixed-Integer Problems. Operations Research, 40(1):109–116, 1992.

9. M. Padberg. The Boolean Quadric Polytope: Some Characteristics, Facets and

Relatives. Mathematical Programming, 45:139–172, 1989.

10. W.B.F. Ryan, S.M. Carbotte, J.O. Coplan, S. O’Hara, A. Melkonian, R. Arko,

R.A. Weissel, V. Ferrini, A. Goodwillie, F. Nitsche, J. Bonczkowski, and R. Zem-

sky. Global Multi-Resolution Topography Synthesis. Geochem. Geophys. Geosyst.,

10(3):Q03014, 2009.