Acta Polytechnica Vol. 52 No. 6/2012

Global Optima for Size Optimization Benchmarks
by Branch and Bound Principles

Adéla Pospíšilová, Matěj Lepš

Faculty of Civil Engineering, CTU in Prague, Thákurova 7, 166 29 Prague, Czech Republic

Corresponding author: adela.pospisilova@fsv.cvut.cz

Abstract
This paper searches for global optima for size optimization benchmarks utilizing a method based on branch and bound
principles. The goal is to demonstrate the process for finding these global optima on the basis of two examples. A suitable
parallelization strategy is used in order to minimize the computational demands. Optima found in the literature are
compared with the optima used in this work.

Keywords: benchmarks, discrete sizing optimization, branch and bound method, global optima, parallel programming.

1 Introduction
Optimization and search methodologies have become
very popular for making products more desirable. The
shape of a structure, the amount of reinforcement,
the cross-sections, sheet thicknesses, design of the
concrete mixture, and many other properties can be
optimized. Recently, many heuristic algorithms have
been developed and tested on benchmarks in order
to assess their performance. In order to compare dif-
ferent optimization methods, it is necessary to have
reliable information on the global optima of the bench-
marks. In the past, it was not possible to obtain
these optima by exhaustive search approaches, due to
the large computational demands. As computational
power is growing year by year, it now seems to be the
right time to deal with this issue.
This paper outlines a process for searching global

optima for sizing discrete optimization benchmarks.
Various optimization methods can be used for obtain-
ing optima, e.g. gradient methods [1], heuristic meth-
ods, and evolutionary algorithms [2]. These methods
do not guarantee that a global optimum is obtained
because only a portion of the space is explored. It is
not always necessary to obtain the global optimum.
A local optimum with good qualities found within
a short time can be considered as a mark of a high
quality algorithm. However, without knowledge of the
global optima for selected benchmarks, it is not possi-
ble to make a reliable assessment of the performance
of the methodology that is used.
In our work, we look for global optima for some

fundamental benchmarks, using a method based on
branch and bound principles. This approach requires
two values called bounds for determining the searched
space. A good estimate of these bounds reduces the
searched space but still ensures that global optima

can be found. The optimization problem for the
benchmarks presented in this paper is defined by an
objective function that is easy to solve, constraints
with high computational demands and with a searched
space that is discrete and huge. The algorithm pre-
sented here can be used for obtaining the global opti-
mum for problems similar to those discussed in this
paper.

2 Sizing optimization

Sizing optimization [3] is a type of structural opti-
mization that deals with truss-like structures. These
structures are defined by a fixed topology, material,
loading, supports, and a set of cross-sections or, al-
ternatively, minimum and maximum cross-sectional
areas of individual truss bars. The objective function
is the weight of the structure or its volume. The objec-
tive function is linear and easy to solve. Constraints
are maximum stresses and maximum displacements,
respectively. These functions are non-convex in the
most cases, and it is more time-consuming to solve
them that to solve the objective function.

The goal is to find sections for a given structure that
satisfy the prescribed constraints and have the lowest
possible weight. The selection of cross-sections from
the given database then defines a discrete optimization
problem, and variables chosen from given limits lead
to a continuous case. The continuous optimization
problem can be efficiently solved by mathematical
programming methods, e.g. gradient-based methods,
as will be shown below. When using discrete variables,
no such option is available. Thus we first give our
attention to the discrete case.

Acta Polytechnica Vol. 52 No. 6/2012

3 Discrete problem
The goal is to find a combination of cross-sections from
the given list of profiles that leads to the lowest pos-
sible weight, while still fulfilling the given constraints.
We present two methods that are able to find global
optima for this discrete optimization problem.

3.1 Enumeration
Enumeration (also called a Brutal Force or Exhaustive
search) is the simplest method for obtaining a global
optimum of the discrete optimization problem. Here,
it is necessary to compute values of an objective func-
tion and constraints for every combination of cross-
sections from a given set. The enumeration therefore
poses very large computational demands. If there are
n sections and k variables (i.e., truss bars or groups of
bars), then nk possible solutions exist, i.e., the prob-
lem grows exponentially with the growing number of
variables. Enumeration can therefore be applied only
for small structures or for analysing the neighborhood
of some local optima.

3.2 A method based on branch and
bound principles

A branch and bound method is another method for
obtaining global optima. Land and Doig [4] invented
this method for linear programming problems. Later,
it was modified for discrete problems and for mixed-
discrete problems [5].

Branch and bound methods are based on a dividing
the main problem into several subproblems, known as
branches. To estimate which branches are to be eval-
uated, the existence of the lower and upper bounds
is assumed to restrict the searched space. The lower
bound can be obtained by any continuous optimiza-
tion method, because the global optimum with dis-
crete design variables will never provide a lower value
of the objective function than the global optimum
with continuous design variables. The upper bound
can be obtained by any heuristic method, because a
local optimum always has a greater or equal value of
the objective function than the global optimum. Since
the constraints for the sizing optimization problem
are more computationally demanding than the value
of the objective function, they are calculated only
for solutions that lie between the lower and upper
bounds. If we obtain a subproblem with a value of
the objective function outside the given bounds, the
rest of the branch is not calculated because a global
optimum cannot be located there. The more accurate
the estimates of the lower and upper bounds are, the
narrower the searched space can be. It is thus advan-
tageous to decrease the upper bound on the basis of
already obtained objective function values.

Although this methodology is more efficient than
enumeration, it is still time consuming. However,
it is possible to parallelize the algorithm to reduce
the computation time. The main idea of the parallel
version of the algorithm is presented in Section 6.

4 Continuous problem
A continuous optimization problem is more complex
than a discrete problem, because an infinite num-
ber of potential solutions exist in the space with real
numbers. Therefore, for non-convex problems, it can-
not be guaranteed that the optimum that is found
is the global optimum. However, powerful and well-
established continuous optimization algorithms such
as mathematical programming methods, can be used
for obtaining a potential global optimum. Obtaining
a potential global optimum with continuous variables
is therefore less demanding than solving the optimiza-
tion problem with discrete variables.
The main disadvantage of this methodology is the

uncertainty of the quality of the solution. This can
be overcome in one of two ways:

• The branch and bound method expects that the
lower bound has the same (or a lower) value of
the objective function as the global optimum with
continuous variables. Since the global optimum
of the continuous problem cannot be generally
known, the true lower bound cannot be ensured.
As a solution, the lower bound is set to its low-
est potential minimum, i.e., without using any
continuous optimization method. This process
provides a real global optimum with discrete vari-
ables. In most cases, however, the searched space
will be enormous for the computation of all pos-
sible solutions in real time.

• Other approaches do not fully guarantee the ac-
quisition of the global optimum. Nevertheless,
the probability of obtaining the global optimum
is acceptable. These approaches are based on
estimating the global optimum with continuous
variables and the value of its objective function.
We use nonlinear programming that is imple-
mented in the MATLAB environment (e.g. the
fmincon function). This routine is executed sev-
eral times from random initial points. If the ob-
tained optima do not differ from each other and
the results are comparable to optima published in
the available literature, the estimate is considered
as credible. If the obtained optima differ from
each other, then it is not possible to use them as
the lower bound. The first approach (without us-
ing the continuous optimization method) is then
used, or the lower bound is estimated to be e.g.
20 % lower than the solution what is obtained.

Acta Polytechnica Vol. 52 No. 6/2012

10 in

3 kips 5 kips

1 kip

1
0

2 3

2
4

x
y

Figure 1: The 5-bar truss.

All continuous optima for problems mentioned be-
low were consistent with published results. For the
sake of certainty, the nonlinear programming method
was launched hundred times with different starting
vectors, and the best solution was considered as the
lower bound.

5 Benchmarks

5.1 5-bar truss
It was necessary to have a representative example of
a structure that was small enough for computational
demands and yet big enough for branching purposes
in order to test the branch and bound algorithm.
Note that imperial units are used throughout the
text, because our solutions will be compared with
published optima in the available literature with the
same units.

The structure in Fig. 1 has four nodes and five truss
bars, and is made from aluminium. The density of
the material is 0.1 lb/in3 and the Young’s modulus
is equal to 104 ksi. The allowable stress is limited to
±60 ksi in each element, and the displacements are
limited to ±0.06 in along the horizontal and vertical
directions. The continuous variables can vary between
the lower bound 0.01 in2 and the upper bound 0.1 in2.

A function for nonlinear programming fmincon [6]
offers four variants of the optimization algorithms.
In this paper, the Active Set Method [7] is suitable
for our continuous sizing optimization problem. It is
based on changing inequalities into equalities, followed
by the line-search algorithm leading to a quadratic
subproblem. This procedure is repeated in a sequence
which converges in the limit to a critical point [8].

A starting point, i.e., a design variable vector com-
posed of cross-sectional areas, an objective function,
constraints and lower and upper bounds of the vari-
ables are necessary as an input at the beginning of
the algorithm. The objective function is the weight

Variable Units Discrete
optimization

Continuous
optimization

E B&B fmincon()

A1 in2 0.05 0.05 0.0500

A2 in2 0.01 0.01 0.01

A3 in2 0.06 0.06 0.0471

A4 in2 0.02 0.02 0.0167

A5 in2 0.01 0.01 0.01

m lb 0.179 0.179 0.157

max |wj| in 0.059 0.059 0.06

max |σi| ksi 59.371 59.371 60.006

wlim in 0.06 0.06 0.06

σlim ksi 60 60 60

Table 1: 5-bar truss optima. B&B — Method based
on Branch and Bound principles, E — Enumeration.

of the structure

m = f(A) = ρ
N∑
i=1

AiLi, (1)

where N = 5 is the number of truss bars, Ai is the
cross-sectional area and Li is the length of element i,
and ρ is the density of the material. Constraints are
defined as inequalities:

max |σi|− 60 ≤ 0, (2)
max |wj|− 0.06 ≤ 0, (3)

where max |σi| is the maximum absolute value of
stresses, j is an ordinal number of independent dis-
placements and max |wj| is the maximum absolute
value of displacements. These values can be obtained
by several methods. In this paper, geometrically and
physically linear behaviour was assumed and the finite
element method was used, see e.g. [9].

The results for the continuous optimization problem
of the 5-bar truss appear in Tab. 1. The objective
function value of the optima obtained with the Active
Set Method is later used as the lower bound for the
branch and bound method. Dealing with inequalities
in the form of equalities is solved by a penalty type
approach, and therefore the exact fulfillment of the
constraints therefore cannot be ensured, see Table
1. This discrepancy is not crucial since only a lower
bound is needed, not the global continuous optimum.
An identical topology of the 5-bar truss is used

for the discrete optimization version. The mate-
rial properties and constraints are also identical.

Acta Polytechnica Vol. 52 No. 6/2012

initialization

set variables to
the smallest values

compute m

compare m with
mmin and mmax

compute constraints
Are they satisfied?

mmin ≤ m < mmax

YES

m < mmax?
YES

mmax = m

increase last
variable by one

compute m

find combination
m > mmin

m < mmin m ≥ mmax

are all variables set to
the highest values?

find combination
m < mmax

END

YES

STEP 1

STEP 2
STEP 3

STEP 4

STEP 5

STEP 6

Figure 2: A flow chart of the algorithm.

The cross-sectional areas are chosen from the set
{0.01, 0.02, . . . , 0.1} in2. Since the structure has five
elements (k = 5) and 10 cross-sectional areas (n = 10),
the number of all possible solutions is nk = 105.
Therefore, the structure is small enough and the dis-
crete global optimum can be obtained by the enumer-
ation method. The results appear in Tab. 1.

The lower bound for the branch and bound method
is set to the optimum objective function value ob-
tained with continuous variables. The upper bound
is set to the estimated weight of 0.23 lb, which is 25 %
higher than the global optimum value of the objective
function obtained by the enumeration. The space is
searched systematically between these two bounds
until the global optimum is found.

A flow chart of the algorithm is depicted in Fig. 2.
The steps can be described as follows:

1. First, we have to decide which values will be used
as the initial point. It is appropriate to begin
with the smallest profiles and then increase them,
since the objective function is linear with respect
to the cross-section areas. From the programming
point of view, it is easier to use integer variables
that are the ordinal numbers of the given set of
cross-sectional areas — set M. For example, the
initial design variable vector is 1 1 1 1 1, which
means that the first area (0.01 in2) from the given
set is attached to each truss bar, according to
the numbering of the trusses shown in Fig. 1.

2. The value of the objective function is then calcu-
lated and compared with the lower mmin and up-
per mmax bounds. If the weight of the structure
is less than mmin, the algorithm will go to Step 3.

If the weight of the structure is between mmin
and mmax, the algorithm proceeds to Step 4. If
the weight of the structure is greater than mmax,
Step 5 is executed.

3. The value of the objective function is less than
mmin. It is therefore necessary to find a combi-
nation of variables that corresponds to weight
larger than mmin. The last variable is raised to
its maximum value, e.g. as 1 1 1 1 10, and the
value of the objective function is calculated and
compared to mmin.

(a) If the value of the objective function is still
less than mmin, the algorithm searches for
a combination with greater weight than the
lower bound. This can be done as follows.
The next-to-last variable is repeatedly raised
by one, e.g. to (1 1 1 2 10). If the value
of the next-to-last variable reaches its maxi-
mum, it is decreased to its minimum and the
third from the end variable value is raised
by one. The algorithm will go to Step 2 at
the moment when all variables are set such
that m > mmin.

(b) If the value of the objective function is
greater than the lower bound, the last vari-
able value is decreased to its minimum
(1 1 1 1 1) and is then increased one by
one (1 1 1 1 2, 1 1 1 1 3, ..., etc.) un-
til the weight is greater than mmin. If
mmin > m the algorithm goes to Step 2.

4. The value of the objective function is greater than
mmin and less than mmax. The global optimum
is located somewhere in this subspace. Therefore,
the constraints are evaluated, i.e the stresses and
displacements are calculated.

(a) If the constraints are fulfilled, i.e.,
max |σi| ≤ 60 ksi and max |wj| ≤ 0.06 in,
the upper bound is updated to the actual
objective function value mmax = m. Thus
the upper bound is pushed down towards
the global optimum and the searched space
is reduced. The last variable value is then
increased by one. If this variable value ex-
ceeds its maximum possible cross-sectional
area value from a given set, e.g. 1 1 5 11
1, its value is set to the minimum possible
value, and the next-to-this variable value
is increased by one, i.e., 1 1 6 1 1. The
algorithm goes to Step 2.

(b) If the constraints are not fulfilled, the value
of the last variable is increased by one. The
algorithm continues with Step 2.

5. The value of the objective function is greater than
mmax. The value of the last variable is decreased

Acta Polytechnica Vol. 52 No. 6/2012

1.41%
6.13%

92.46%

number of potential solutions below the lower
bound (step 3)

number of potential solutions between the lower
bound and the upper bound (step 4)

number of potential solutions above the upper
bound (step 5)

Figure 3: A pie chart of a distribution of the 5-bar
truss problem solutions solved by the branch and
bound method.

to its minimum, the next-to-last variable value
is increased by one and the objective function
value is calculated. If the variable exceeds its
maximum possible value from the given set, the
algorithm acts as in Step 4a.

(a) If the value of objective function m is lower
than mmax, the algorithm goes to Step 2.

(b) If objective function value m is greater than
mmax, the value of the third variable from
the end increases by one and the value of
the next-to-last variable is set to its mini-
mum. The algorithm continues in this way
until the objective function value is less than
mmax. If there is no such combination of
cross-sectional areas, the algorithm is termi-
nated.

6. If all variable values are set to their maxima, the
algorithm ends.

Fig. 3 shows the distribution of 5-bar truss potential
solutions. The dark grey part shows the number of
potential solutions below the lower bound, where only
the objective function values are calculated, i.e., Step 3
of the algorithm. The grey part shows the number of
potential solutions between the lower and the upper
bound, where the values of the objective function
and also the constraints are calculated (Step 4). The
global optimum is included in this subspace. The light
grey part represents the number of potential solutions
above the upper bound, where only the objective
function values are calculated (Step 5). It is obvious
that there is no need to compute constraints for more
than 90% of potential solutions. Since the evaluation
of constraints is very computationally demanding, this
results in significant time savings. The reason for the
bigger number of potential solutions above the upper

3 5

1
0
0

y
200200

[in]

Figure 4: A 25-bar truss.

bound is that the optimum is placed relatively close
to the lower bound in the searched subspace.

Tab. 1 presents results for the continuous optimiza-
tion problem along with results for the discrete prob-
lem solved by the enumeration and the method based
on branch and bound principles. Since the enumer-
ation calculates the values of the objective function
as well as constraints for all potential solutions, it is
not possible to omit the global optimum. The results
obtained by both presented methods are identical
and this comparison serves as the verification of the
branch and bound method.

5.2 25-bar truss
The 25-bar truss is one of the most widely-used bench-
marks for size optimization. It was introduced by Fox
and Schmit [10] in 1966. The structure has ten nodes
and four supports (at nodes 7–10), see Fig. 4; there-
fore there are 18 free displacements. The structure is
symmetric so some elements were gathered to eight
groups, listed in Tab. 2. The truss is made of alu-
minium material, with density equal to 0.1 lb/in3 and
with the Young’s modulus equal to 104 ksi. The load-
ing is defined in Tab. 3. All cross-sectional areas are
chosen from the given set: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6,
0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9,
2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2 and 3.4 in2,
see [11]. The continuous variables range from 0.1 in2
to 3.4 in2. The allowable stress is set to ±40 ksi in all
truss bars and the maximum allowable displacement
is ±0.35 in at all nodes along the x, y and z directions.
The results for the continuous case are shown in

Tab. 4 and they are compared with the results pub-
lished in the available literature. The discrete case
cannot be enumerated within a reasonable time be-
cause the number of potential solutions is nk = 308 =
6.561 · 1011, where k is the number of element groups.
For the discrete case, the lower bound was set to the

Acta Polytechnica Vol. 52 No. 6/2012

Group of bars Conectivities

A1 1-2

A2 1-4, 2-3, 1-5, 2-6

A3 2-5, 2-4, 1-3, 1-6

A4 3-6, 4-5

A5 2-4, 5-6

A6 3-10, 6-7, 4-9, 5-8

A7 3-8, 4-7, 6-9, 5-10

A8 3-7, 4-8, 5-9, 6-10

Table 2: Member grouping for the 25-bar truss.

Node Fx Fy Fz

1 1.0 −10.0 −10.0

2 0 −10.0 −10.0

3 0.5 0 0

6 0.6 0 0

Table 3: Loadings for the 25-bar truss (kips).

Variable Unit Perez &
Behdinan

This paper

A1 in2 0.1 0.1

A2 in2 0.457 0.421

A3 in2 3.4 3.4

A4 in2 0.1 0.1

A5 in2 1.937 1.917

A6 in2 0.965 0.966

A7 in2 0.442 0.471

A8 in2 3.4 3.4

m lb 483.84 483.82

max |σi| ksi 6.15 6.13

max |wj| in 0.35 0.35

σlim ksi 40 40

wlim in 0.35 0.35

Table 4: Comparison of results for the 25-bar truss
continuous case.

1
2
3
4
5
6
7
8

1 2 3 4 5 6 7 8

sp
ee

d-
up

number of cores

speed-up
linear speed-up

Figure 5: A speed-up of the 25-bar truss problem
solved by the parallel branch and bound method.

value found with continuous optimization, and the
upper bound was set to the worst available solution
in the literature [12].

6 Parallelization
The 25-bar truss is relatively computationally de-
manding. Since the evaluations of the solutions are
independent from each other (except updating the
upper bound mmax, as described in Step 4a of the
algorithm), the method can be run in a parallel way.
Nowadays, modern computers are equipped with sev-
eral core processors, and we can to make use of this
computational power. The MATLAB environment
offers several parallelization tools, but not all of them
provide shared memory. This consideration is essen-
tial for our algorithm for updating the upper bound.
The appropriate method is the spmd method, i.e.,

the Single Programm Multiple Data method, see
e.g. [13] for more details. The spmd statement sepa-
rates the block of the code to be run simultaneously
on multiple labs. As in the parfor loop method,
the matlabpool open N command opens the required
number of labs. Data can be sent to another lab by
the labSend(data, X) command, where X is the in-
dex of the receiving lab to which the data is sent. The
data is received by the labReceive(Y) command,
where Y is the index of the lab from which the data
will come. It is appropriate to split the data only at
one so-called master lab and to receive data with the
other so-called slaves. The master can also process
its own data as well.
The main problem here is to estimate a proper

amount of data for each lab. If the data is sent too
often communication between master and slaves is too
costly. In the 25-bar truss task, permutations with
repetition are generated in advance for several groups
of elements (e.g. four groups), and the remaining
groups of elements (four other groups) are generated
in the branch and bound method independently at
each lab. The generated combinations are assigned
to individual labs in advance, and then the algorithm
continues in the same way as for the 5-bar truss task.
The maximum mmax values are collected at the end

Acta Polytechnica Vol. 52 No. 6/2012

480
500
520
540
560

1 10 100 1000 10000

w
ei

gh
t

[lb
]

iterations of the algorithm

Figure 6: A graph of the decreasing upper bound
for the 25-bar truss problem.

of each iteration. The smallest one is chosen as a new
mmax and is resent to every lab as an initial value
of mmax for another iteration. If all data has been
used, the smallest value of mmax is taken as the global
optimum.
For the parallel version of the algorithm, scaling

the algorithm is very important. If the algorithm
is scaled badly, parallelization is not useful at all.
Ideally, we would like to achieve linear scaling, i.e.,
speed-up of n on n cores. However, it is very hard to
obtain linear scaling, e.g. because of the time spent
on communications. Fig. 5 shows a graph where the
speed-up of the parallel algorithm is compared at 1 to
8 labs1. The HP Xeon Z600 Workstation with two In-
tel Xeon E5520 4-cores processors, frequency 2.27 GHz
was used for computations within the Matlab R2009a
64-bit Debian GNU/Linux environment.

7 Conclusions
Fig. 6 shows a graph with the decreasing upper bound
mmax for the 25-bar truss problem. The value of mmax
determines the best-so-far solution found during the
whole algorithm run. It can be interpreted as the
convergence of the objective function to the global
optimum. All combinations generated in advance
were divided into smaller blocks of data containing
fifty combinations and these “packs” were sequentially
sent to eight labs. The number of all iterations was
therefore 304/(8 · 50) = 2025. The global optimum
was gained in the 66th iteration. Nevertheless, it
should be pointed out that the number of iterations
depends strictly on the data ordering or the starting
point (minimum vs maximum cross-sectional areas).
Since the task is to find the global optima, the whole
subspace of potential solutions must be searched for,
and it is not possible to shorten the computation.
In Tab. 5, we compare the optima obtained with

the branch and bound method with the heuristic
algorithms found in the literature. The result of the
branch and bound method (B&B) is identical to the

1It was not necessary to compute the whole task. Some vari-
ables were fixed to prescribed values, here 2 out of 8 variables,
and the algorithm was run with this restriction.

solution presented by Kripka [15] using the Simulated
Annealing method (SA). However, he did not search
the whole subspace of possible solutions, so he could
not be sure that the obtained optimum is the global
one.
To the best of the authors’ knowledge, global op-

tima for computationally demanding tasks such as
the 25-bar truss problem have not been published
yet. We hope that by publishing the algorithm as
well as the value of the global optimum we will intro-
duce a quality standard that will help to improve new
optimization methods.

Acknowledgements

This work was supported by the Grant Agency
of the Czech Technical University in Prague,
grants number SGS11/021/OHK1/1T/11 and number
SGS12/027/OHK1/1T/11, the Czech Science Foun-
dation GACR, grant number P105/12/1146 and by
Ministry of Education, Youth and Sports, grant num-
ber MSM 6840770003.

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Acta Polytechnica Vol. 52 No. 6/2012

Variable Units This
paper

Kripka Lemonge &
Barbosa

Li &
Liu

Wu &
Chow

Coello Rajeev &
Krishnamoorthy

[15] [16] [17] [11] [18] [12]

B&B SA GA PSO GA GA GA

Date 2012 2004 2004 2009 1995 1994 1992

A1 in2 0.1 0.1 0.1 0.1 0.1 1.5 0.1

A2 in2 0.4 0.4 0.3 0.3 0.5 0.7 1.8

A3 in2 3.4 3.4 3.4 3.4 3.4 3.4 2.3

A4 in2 0.1 0.1 0.1 0.1 0.1 0.7 0.2

A5 in2 2.2 2.2 2.1 2.1 1.5 0.4 0.1

A6 in2 1 1 1 1 0.9 0.7 0.8

A7 in2 0.4 0.4 0.5 0.5 0.6 1.5 1.8

A8 in2 3.4 3.4 3.4 3.4 3.4 3.2 3

m lb 484.33 484.33 484.85 484.85 486.29 539.78 546.01

max |σi| ksi 6.20 6.20 6.11 6.11 6.01 6.66 6.77

max |wj| in 0.35 0.35 0.35 0.35 0.35 0.34 0.35

Table 5: A comparison of results for the 25-bar truss, discrete case, from the literature and from our work.
B&B — Method based on Branch and Bound principles, SA — Simulated Annealing, GA — Genetic Algorithm,
PSO — Particle Swarm Optimization. σlim = 40 ksi, wlim = 0.35 in.

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