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A survey of non-gradient optimization methods in structural engineering

Warren Hare

a

, Julie Nutini

b

, Solomon Tesfamariam

c,

⇑

a

Mathematics, University of British Columbia, Kelowna, BC, Canada

b

Computer Science, University of British Columbia, Vancouver, BC, Canada

c

School of Engineering, University of British Columbia, Kelowna, BC, Canada

article info

Article history:

Received 8 November 2012

Received in revised form 11 March 2013

Accepted 11 March 2013

Keywords:

Optimization

Structural engineering

Non-gradient methods

Heuristic methods

Swarm methods

Derivative-free optimization

abstract

In this paper, we present a review on non-gradient optimization methods with applications to structural

engineering. Due to their versatility, there is a large use of heuristic methods of optimization in structural

engineering. However, heuristic methods do not guarantee convergence to (locally) optimal solutions. As

such, recently, there has been an increasing use of derivative-free optimization techniques that guarantee

optimality. For each method, we provide a pseudo code and list of references with structural engineering

applications. Strengths and limitations of each technique are discussed. We conclude with some remarks

on the value of using methods customized for a desired application.

Ó2013 Elsevier Ltd. All rights reserved.

1. Introduction

Optimization is the process of minimizing or maximizing an

objective function (e.g. cost, weight). Three main types of optimi-

zation problems that arise in structural engineering are [1,2]: siz-

ing optimization, shape optimization, and topology optimization.

Sizing optimization entails determining the member area of each

element. Shape optimization entails optimizing the proﬁle/shape

of the structure. Topology optimization is associated with connec-

tivity of structural elements. Traditionally, the three optimization

problems were solved independently (e.g., [3]), however, recent

trend shows simultaneous optimization of sizing, shape and topol-

ogy provides better results [2,4].

In this paper, we consider optimization problems of the form

minimize

x

fðxÞ

subject to cðxÞ60

l6x6u;

ð1Þ

where f:R

n

!R;cðxÞ¼ðc

1

ðxÞ;...;c

m

ðxÞÞ and 6should be inter-

preted coordinate-wise. We permit l

j

,u

j

=±1,j2{1, ...,n}, to allow

for the possibility of unbounded variables.

If the objective function of an optimization problem is

smooth (i.e., differentiable) and gradient information is reliable,

then gradient based optimization algorithms present an extre-

mely powerful collection of tools for solving the problem. How-

ever, in some structural engineering problems, such as when

simulations are employed to imitate problem conditions, gradi-

ent information may not be available for the problem. Even if

gradient information is available, it can be unreliable or difﬁcult

to compute. Thus, non-gradient methods are incredibly useful

optimization tools.

As the name suggests, non-gradient methods do not require

gradient information to converge to a solution. Rather, these meth-

ods solely use function evaluations of the objective function to

converge to a solution. We note that if gradient information is

available for a well-behaved problem, then a gradient based meth-

od should be used. However, when gradient information is not

available, non-gradient methods are practical alternatives. Several

reviews of non-gradient methods for optimization problems in

structural engineering have been published. The majority of these

focus on heuristic methods. In 1991, a review of genetic algorithms

for structural optimization was published [5]. In 2002, a more gen-

eral review of evolutionary algorithms for structural optimization

was published [6]. In 2007, a review focused on the design of steel

frames via stochastic search methods was published [7]. In 2008, a

review on the use of simulated annealing methods for structural

optimization [8] and a general review on publications of structural

engineering applications using particle swarm optimization [9]

were published. In 2009, a review on the use of the harmony

search methods in structural design optimization was published

0965-9978/$ - see front matter Ó2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.advengsoft.2013.03.001

⇑

Corresponding author. Tel.: +1 250 807 8185.

E-mail address: Solomon.Tesfamariam@ubc.ca (S. Tesfamariam).

Advances in Engineering Software 59 (2013) 19–28

Contents lists available at SciVerse ScienceDirect

Advances in Engineering Software

journal homepage: www.elsevier.com/locate/advengsoft

Author's personal copy

[10]. In 2011, a review focused on the design of skeletal structures

using a variety of heuristic techniques was published [11]. Most re-

cently, in 2012, a comprehensive review of stochastic search heu-

ristics was published [12].

In this paper, we present a detailed review of the non-gradient

methods for structural optimization. We also provide a list of ref-

erences that utilize the optimization methods. We include the

methods in the review papers previously mentioned, as well as

several other methods. We also include the more recent Deriva-

tive-free Optimization (DFO) methods that have become increas-

ingly popular in optimization applications. Unlike the general

category of non-gradient methods, DFO methods are supported

by mathematical convergence theories, which ensures that the

algorithms converge to a local minimizer of the objective function.

Due to their practical utility and the numerous problems suited

to them, new non-gradient algorithms are frequently developed;

for example, the very recent magnetic charged system search

[13], which adapts the (also recent) charged system search [14].

In this paper, we choose to focus on methods that appear fre-

quently in the literature. Thus, we exclude some recently proposed

methods. However, we emphasize that new non-gradient methods

are regularly providing improved solutions to many structural

engineering problems.

The remainder of this paper is organized as follows. In Section 2,

we present some of the most popular heuristic methods used in

structural engineering: evolutionary algorithms. These methods

use techniques that imitate natural evolution. In Section 3,we

present some heuristic methods inspired by physical processes

and the nature of stochastic processes. In Section 4, we present

heuristic methods inspired by self-organizing systems. These

methods are often referred to as swarm algorithms, as they are of-

ten inspired by how animal swarms employ simple rules to devel-

op favorable system behavior. In Section 5, we present formalized

methods that are strengthened by mathematical convergence the-

ory. We refer to these methods as Derivative-free Optimization, as

is commonly used in the mathematical community. Each of these

sections is broken into several subsections that describe examples

of speciﬁc algorithms. In Section 6, we consider our observations

from the previous sections, present our conclusions and provide

a summary table of the methods discussed.

2. Evolutionary algorithms

Evolutionary algorithms are a class of non-gradient population-

based algorithms used in many areas of engineering optimization.

These methods use techniques that imitate natural evolution. They

follow the four general steps of reproduction, mutation, recombi-

nation and selection and use a ﬁtness function to determine the

conditions that support survival.

2.1. Genetic algorithm

A Genetic Algorithm (GA) is probably the most commonly used

evolutionary algorithm and one of the more common non-gradient

methods. These methods were originally proposed by John Holland

in 1975 [15]. A GA selects an initial population of potential solu-

tions, say P(t), for each iteration tto the problem at hand. Using

stochastic transformations, some solutions will undergo a muta-

tion or crossover step. These new potential solutions are referred

to as the offspring, say C(t). From both P(t) and C(t), the ‘most ﬁt’

solutions (solutions with the better objective values) are selected

to form a new population, P(t+ 1). After the evaluation of several

generations, the algorithm hopefully converges to the optimal or

sub-optimal solution of the objective function. Generally, the

structure of the GA is as follows [16]:

procedure GeneticAlgorithm

begin

Initialize and evaluate P(t);

while (not termination condition) do

begin

Recombine P(t) to yield C(t);

Evaluate C(t);

Select P(t+ 1) from P(t) and C(t);

end

end

GAs have been applied to numerous structural engineering

applications: structural reliability [17,18], bridge design, structure,

maintenance and repair [19–21], design of welded steel plate gir-

der bridges [22], seismic zoning [23], seismic design of lifeline sys-

tems [24], truss structure optimization [25–27], the size, shape and

topology of skeletal structures [28] and design optimization of

steel structures [29–32], reinforced concrete ﬂat slab buildings

[33], steel telecommunication poles [34] and viscous dampers

[35]. See also [36–49,31,50–58].

2.2. Evolutionary strategies

Evolutionary strategies (ES) are a subclass of evolutionary algo-

rithms. The structure of these methods was originally developed

by Rechenberg and Schwefel ([59–62]). To start, an ES deﬁnes an

initial parent population of size

l

that consists of potential solu-

tions, say B

ð0Þ

p

, to the problem at hand. Each individual a

k

in B

ð0Þ

p

is

comprised of a parameter set y

k

, its objective value F

k

:¼F(y

k

)

and an evolvable set of strategy parameters s

k

. Next, the parent

population reproduces, generating koffspring, where kis a ﬁxed

parameter of the method. To do this, ﬁrst there is a marriage step,

where one family Cof size

q

is randomly chosen from the parent

population at time t;B

ðtÞ

p

. Then for the individuals in family C, their

strategy and object parameters are recombined. These new param-

eters are then mutated, forming the offspring population at time

t;B

ðtÞ

o

. Finally, the selection step forms a new parent population

B

ðgþ1Þ

p

. There are two main types of ESs for different numbers of par-

ents and offspring, namely (

l

+k)-ES and (

l

,k)-ES. Generally, the

structure of the ES is as follows ([63]):

procedure EvoluntionStrategy

begin

Initialize parent population B

ð0Þ

p

;

while (not termination condition) do

for n=1:kdo begin

C

n

= MarriageðB

ðtÞ

p

;

q

Þ;

s

n

= s_recombination (C

n

);

y

n

= y_recombination (C

n

);

~

s

n

¼s_mutation (s

n

);

~

y

n

¼y_mutation ðy

n

;~

s

n

Þ;

end

Update B

ðtÞ

o

;

Perform selection and update parent population B

ðtÞ

p

;

end

end

Evolutionary strategies appear in the literature often and have

been applied to several structural engineering problems: optimiz-

ing truss structures ([64–66]), optimizing a connection rod shape

and minimizing the volume of a square plate with a central cut-

out ([67]), and the design of a cantilever beam ([68]), steel frames

([30]) and cylindrical shells ([69]).

20 W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28

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2.3. Strengths and limitations of evolutionary algorithms

As a class of heuristic algorithms, there is no mathematical con-

vergence theory for evolutionary algorithms – and thus no assur-

ance of optimality of the ﬁnal solution found. However, in

practice, evolutionary algorithms can be very successful in ﬁnding

good solutions quickly. The number of function evaluations evolu-

tionary algorithms require can be scaled independent of the

dimension of the problem, making them versatile for large scale

problems. However, do to the need to maintain large populations

of candidate solutions, evolutionary algorithms can be cumber-

some for small scale problems. Most papers that compare the per-

formance of non-gradient methods on one application include at

least one comparison to an evolutionary algorithm (see Table 1).

evolutionary algorithms are often considered a baseline approach,

and the method to beat if you wish to claim a new algorithm is of

high quality.

3. Physical algorithms

3.1. Harmony search

The Harmony Search (HS) algorithm was ﬁrst introduced by

Geem, Kim and Loganathan in 2001 [70]. As the name suggests, this

algorithm mimics the evolution of a harmony relationship between

several sound waves of differing frequencies when played simulta-

neously. In music, a best state (aesthetically pleasing harmony) is

desired; in optimization, the ‘best state’ is achieved at the global

optimum. The processes of random selection, memory consider-

ation and pitch adjustment are all incorporated in this algorithm.

There are two main parameters used in the HS algorithm: the Har-

mony Memory (HM) accepting rate, denoted by r

accept

, and the

pitch adjustment rate, denoted by r

pa

. As their names suggest,

r

accept

is the rate at which a new harmony is accepted into the

HM, and r

pa

controls the degree to which the pitch can be adjusted.

The basic structure of an HS algorithm is as follows [71]:

procedure HarmonySearch

begin

Initialize parameters, including r

accept

and r

pa

;

Generate initial HM with random harmonies;

while t< max number of iterations

while i6number of variables

if random value rand <r

accept

Choose value from HM for the variable i;

if rand <r

pa

Adjust the value by adding certain amount;

end if

else

Choose a new rand value;

end if

end while

Accept the new harmony if better;

end while

Find the current best harmony (solution);

end

Several examples of structural engineering problems that have

been solved using an HS algorithm are: optimization of truss

structures [50,72], optimization of pin connected structures

[73], minimum cost design of steel frames [41], and optimum de-

sign of steel frames [40,30,45], steel sway frames [57], cellular

beams [74] and reinforced concrete frames [75]. Several struc-

tural design optimization problems are tackled using an HS algo-

rithm, including sizing and conﬁguration for a truss structure,

pressure vessel design, and welded beam design in [51]. See also

[44,46,31,76,77].

3.2. Simulated annealing

A Simulated Annealing (SA) algorithm [78,79] is a probabilistic

heuristic that mimics the annealing process used in materials sci-

ence. During this process, a material is heated to high tempera-

tures, causing atoms to move from their initial positions and

randomly move through higher energy states. As the temperature

of the material is slowly lowered, the atoms settle into a new con-

ﬁguration that hopefully has a lower internal energy. Translating

this process to an optimization problem, the initial state can be

thought of as a local minimum. The heating of the material trans-

lates to replacing the current solution(s) with a new random solu-

tion(s). The new solution(s) may be accepted according to a

probability based on the resulting function value decrease and on

a ‘temperature’ measure, which slowly decreases as iterations con-

tinue. The temperature parameter allows for solutions to be ac-

cepted that may have a higher objective value, thus avoiding

local minima. The basic structure of a SA algorithm is as follows

[80]:

procedure SimulatedAnnealing

begin

Select an initial state i2Sand initial temperature T>0;

Set temperature change counter t:¼0;

while (bf not termination condition) do

begin

Set repetition counter n:¼0;

repeat

Generate state j, a randomly chosen neighbor of i;

Calculate d=f(j)f(i);

if d< 0 then i j

else if random (0,1) < exp (d/T), then i j;

n n+1;

until n=N(t);

t t+1;

T T(t);

end

end

SA algorithms for structural engineering have been used mainly

in design optimization. Some examples are optimizing tensegrity

systems [81] and the design optimization of truss structures [26],

laminated composite structures [82,36], steel frames [83,30],

cross-sections [84] and concrete frames [85]. See also

[86,42,77,58].

3.3. Ray optimization

Inspired by laws that govern the transition of a light ray from

one medium to another, Ray Optimization (RO) is relatively new

to structural engineering. The algorithm employs a number of

agents that search the space. Agents can be thought of as a par-

ticle of light, with a location and direction. At each iteration,

each agent computes an ‘origin’, which is a point deﬁned by

the average of the best known solution and the best known

solution for the individual agent. Using Snell’s refraction law

and a small random perturbation, each agent’s direction is then

adjusted to move towards the ‘origin’ and the agents location is

updated by moving in the new direction. The basic structure of

RO is as follows [87]:

W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28 21

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procedure RayOptimization

begin

Generate initial conditions for agents i;

while (bf not termination condition) do

begin

if an agent violates boundary, then ﬁx position;

Evaluate objective for each agent;

Determine the so-far best global solution g;

For each agent, determine the so-far best position and

store as local best b

i

;

Check stopping conditions;

Compute origin O

i

for each agent: O

i

¼

1

2

ðb

i

þgÞ;

Apply Snell’s refraction law and random perturbation to

determine

each agent’s movement towards their origin;

end

end

Ray optimization has been successfully applied to spring design,

welded beam design, and truss design [87,88].

3.4. Tabu search

The Tabu Search (TS) method, formally proposed by Glover in

1989 [89], is a local search heuristic that works with other algo-

rithms to overcome the restrictions of local optimality. It is applied

to constrained combinatorial optimization problems that are dis-

crete in nature.

To describe the process of a tabu search, we select an initial solu-

tion x2X,whereXis the feasible set. We let S(x)bethesetofmoves

that move xto an adjacent extreme point. Let T#S,whereTis the

set of tabu moves.ThesetTis determined by a function that employs

previous information from the search process up to titerations prior

to the current iteration. To determine membership in T, there may be

an itemized list or a set of tabu conditions, i.e.,

TðxÞ¼fs2S:sviolates the tabu conditionsg:

As a pseudocode, the TS method has the following form [89]:

procedure TabuSearch

begin

Select an initial x2X;

x

⁄

:¼x,T:¼;,k 0;

begin

if S(x)Tis empty

stop;

else

Set k k+1;

Select s

k

2S(x)Tsuch that s

k

(x) = OPTIMUM (s(x):

s2S(x)T));

end

Let x:¼s

k

(x);

if f(x)<f(x

⁄

)

x

⁄

:¼x;

end

Check stopping conditions;

Update T;

end

end

The TS allows an algorithm to store past information and uses it

to improve the steps taken; it can prevent an algorithm from con-

verging back to a local optimum. In structural engineering, it has

been used to optimize the structural weight of frames [90], to opti-

mize the design of steel structures [83,30] and truss structures

[26], and to evaluate the seismic performance of optimized frame

structures [91]. See also [45].

3.5. Strengths and limitations of physical and stochastic algorithms

Like evolutionary algorithms, there is no mathematical conver-

gence theory for physical and stochastic algorithms. These are de-

signed to break free from local minimizers and have often been

found successful in acquiring better global solutions than other

algorithms. However, the tendency to leave local minimizers can

cause difﬁcultly in convergence. As such, physical and stochastic

algorithms are often used in conjunction with other algorithms

that are designed to zoom in on local solutions. As stochastic based

methods, it is difﬁcult to reproduce results from such algorithms.

Running the same algorithm, on the same problem, may result in

widely different answers. It has be argued that this contradicts

the scientiﬁc desire of reproducibility of experiments. This can be

overcome by careful programming and retention of the ‘random’

strings used.

4. Swarm algorithms

Swarm algorithms imitate the processes of decentralized, self-

organized systems, which can be either natural or artiﬁcial in nat-

ure. The most commonly used swarm algorithms in structural

engineering model biological systems that use simple rules, which

result in the development of an ‘intelligent’ system behavior. The

following swarm algorithms will be discussed in the subsequent

sub-sections: ant colony optimization, particle swarm optimiza-

tion, shufﬂed frog-leaping, and artiﬁcial bee colony.

4.1. Ant colony optimization

As the name suggests, an Ant Colony Optimization (ACO) algo-

rithm follows the processes of an ant colony searching for food.

This algorithm, is a stochastic combinatorial optimization method

that uses mathematical principles from graph theory. Basically, it

models the process of ant foraging by pheromone communication

through path formation. A detailed description of the Ant System,

as originally named by Dorigo, Maniezzo and Colorni, can be found

in [92].

To discuss ACO, it is necessary to deﬁne some terms from graph

theory. A mathematical graph can be though of as a collection of

dots connected by a series of lines. Mathematically, the dots are

called nodes (or vertices) and represented by an index i. A lines con-

necting node ito node jis called an edge and represented by a pair

of indices (i,j). A path is a way of getting from one node to another

node by traveling along the edges.

In ACO, the optimization problem is formulated in terms of

determining the shortest path on graph. In brief, the positions for

each ant are selected and the pheromone trail intensities at itera-

tion t= 0, denoted by

s

ij

(t), for each edge (i,j) are initialized. There-

after, every ant moves from their current position to another

position based on a probability function, which is a function of

two ‘desirability measures’: pheromone trail intensities and visibil-

ity [92]. For a trail travelled by many ants, the pheromone intensity

will be strong, indicating a favorable path. The visibility measure

favors proximity of positions, making closer positions more

desirable.

After a set number of iterations, all ants will have completed a

tour of positions and a measure of the change in pheromone trail

intensities will be updated. This cycle will continue until either a

maximum number of cycles has been completed or every ant fol-

22 W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28

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lowed the same tour. The general structure of an ACO algorithm is

as follows [92]:

procedure AntColony

begin

For every edge (i,j), initialize

s

ij

(t);

Place mants on nnodes;

while (not termination condition) do

begin

for s=1,...,n1

for k=1,...,m

Move kth ant to node jusing probability function

p

k

ij

ðtÞ;

end

end

Move each ant to corresponding starting node;

Calculate each path length of tour, L

k

;

Update shortest path;

Update pheromone trail;

end

end

There are three processes that account for the successful nature

of this algorithm: positive feedback, distributed computation, and

a constructive greedy heuristic. Positive feedback is used as a

search and optimization tool. If a choice is made between different

‘path’ options and the result is good, then that choice will be more

favorable in the future. This results in the quick discovery of good

solutions. Distributed computation mimics the increased effective-

ness of a search carried out by a population of ants working

cooperatively together compared to the same number of ants

working individually. By incorporating this idea into the algorithm,

premature convergence is avoided. A greedy heuristic, i.e., only lo-

cally optimal moves are allowed, is used to ensure that reasonable

solutions are found early on in the search process.

Some structural engineering applications that ACO has been ap-

plied to include optimizing bridge deck rehabilitation [20], mini-

mum weight and compliance problems in structural topology

design [93] and design optimization of truss structures [72], con-

crete frames [75] and steel frames [39,30,31,49,94]. See also

[40,45,46,76,52,58].

4.2. Particle swarm optimization

Particle Swarm Optimization (PSO) algorithms mimic animal

ﬂocking behaviors. These algorithms, originally accredited to Eber-

hart, Kennedy and Shi [95,96], have a similar stochastic nature to

GAs and like GAs, work with a set of potential solutions and the

concept of ‘ﬁtness’. Essentially, particles (candidate solutions)

move around the search space, iteratively improving their ﬁtness

value according to a given quality measure. Each particle is inﬂu-

enced by its neighbor. Simple mathematical formulas for position

x

id

and velocity

v

id

are used to move the iparticles through the

d-dimensional hyperspace, accelerating towards ‘better’ solutions

pbest

i

. For a detailed description of PSO algorithms, see [95]. The

general structure of a PSO algorithm is as follows [97]:

procedure ParticleSwarm

begin

Initialize x

id

,

v

id

and pbest

i

for each i;

while (not termination condition) do

begin

for each i

Evaluate f(x

i

);

Update pbest

i

;

end

for each i

Set gequal to index of neighbor with best pbest

i

;

Use gto calculate

v

id

;

Update x

id

=x

id

+

v

id

;

Evaluate f(x

i

);

if f(x

i

)<pbest

i

Update pbest

i

;

end

end

end

end

PSO algorithms have been applied to several structural engi-

neering problems, such as the optimization of a transport aircraft

wing [98], optimizing bridge deck rehabilitation [20], optimization

of pin connected structures [73], structural damage identiﬁcation

[86,47], continuum structural topology design [99] and optimum

design of reinforced concrete frames [75], cellular beams [74], steel

structures [30] and truss-structures [43,54,72,46,52]. See also

[37,44,45,76,53,77,58].

4.3. Shufﬂed frog-leaping

The Shufﬂed Frog-Leaping (SFL) method is a local search heuris-

tic proposed by Eusuff, Lansey and Pasha in 2006 [100]. It is of the

recent group of evolutionary memetic algorithms. A memetic algo-

rithm, like other swarm algorithms, is a population based approach

inﬂuenced by natural memetics. As the name suggests, the SFL

method mimics the actions of frogs in a swamp. Each stone is a

discrete location and the frogs are trying to ﬁnd the stone with

the largest food source. The frogs are allowed to communicate with

other frogs to improve their position.

Basically, the SFL algorithm allows for the separate evolution of

communities, and then shufﬂes these communities. The shufﬂing

process results in local search information being exchanged be-

tween communities. This exchange of information helps the algo-

rithm move towards a global optimum. In general, the global

exploration SFL algorithm has the following form [100]:

procedure SufﬂedFrogLeaping

begin

Initialize number of memeplexes mand frogs in each

memeplex n;

Sample F=mn virtual frogs U(1), ...,U(F);

Compute performance value f(i) for each frog U(i);

Sort frogs in order of decreasing performance, store in array

X;

Set P

X

equal to best frog’s position;

while (not termination condition) do

begin

Partition frogs into memplexes Y

1

,...,Y

m

(nfrogs in each)

according to

Y

k

=[U(j)

k

,f(j)

k

jU(j)

k

=U(k+m(j1)), f(j)

k

=f(k+m(j1)),

j=1,...,n];

Memetic evolution within each memeplex (for details of

local exploration, see [100]);

Replace Y

1

,...,Y

m

into Xin order of decreasing

performance;

Update P

X

;

end

end

W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28 23

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SFL methods have been applied to the optimization of pipe sizes

for water distribution network design [42] and bridge deck repairs

[20,21].

4.4. Artiﬁcial bee colony

The Artiﬁcial Bee Colony (ABC) algorithm, proposed by Karabog-

a in 2005 [101], follows the food foraging behavior of honey bee

swarms. There are three groups of bees in the model: the scout

bees that ﬂy randomly in the search space; the employed bees that

select a random solution to be perturbed based on the exploitation

of the neighborhood of their food sources; and the onlooker bees

are placed on each food source according to a probability based

selection process [102]. The algorithm is based on the amount of

nectar at each of the nfood sources, with onlookers having prefer-

ence to food sources with high probability values. If a new source

has a higher nectar amount than a source in their memory, then

the new position is updated and the previous position is forgotten.

If a predetermined number of trials controlled by the parameter

limit shows no improvement to a solution, then the food source

is abandoned, and the corresponding employed bee becomes a

scout bee. The general structure of the ABC algorithm is as follows

[102]:

procedure ArtiﬁcialBeeColony

begin

Initialize n,limit, food positions x

i

for i=1,...,neach with

dimension d;

Evaluate the ﬁtness of each food position;

while (not termination condition) do

begin

Employed phase:

Produce new solutions with k2{1, ...,n}, j2{1, ...,d},

/2[0, 1] at random according to

v

ij

=x

ij

+/

ij

(x

ij

x

kj

);

Evaluate solutions;

Apply greedy selection for employed bees;

Onlooker phase:

Calculate probability values for each solution x

i

according to P

i

¼

f

i

P

n

j¼i

f

j

;

Produce new solutions from x

i

selected using P

i

;

Evaluate these solutions;

Apply greedy selection for onlooker bees;

Scout phase:

Find abandoned solution:

if limit exceeds

Replace with new random solution;

end

Update best solution;

end

end

For an introduction and references to the different bee optimi-

zation methods, see the introduction of [77] (Artiﬁcial Bee Colony

Algorithm). The ABC algorithm described above has been applied

to structural optimization problems involving truss structures

[44,77,58], laminated composite components [53], inverse analysis

of dam-foundation systems [48] and welded beam and coil spring

design [55].

4.5. Strengths and limitations of swarm algorithms

Like other heuristic methods, there is no mathematical con-

vergence theory for swarm algorithms. Swarm algorithms are

often designed with very speciﬁc problems in mind, and as a

result may be ineffective on problems with different structures.

However, when applied to the speciﬁc problem they are de-

signed for, swarm algorithms have be found to be highly

effective.

5. Direct search methods

The research area of Derivative-free Optimization has blos-

somed in recent years. As previously stated, these methods do

not require derivative information and have mathematical conver-

gence theory. The following Derivative-free Optimization algo-

rithms will be discussed in the subsequent sub-sections:

directional direct search, simplicial direct search, simplex gradient

methods and trust region methods.

5.1. Directional direct search

In Directional Direct Search (DDS) methods, a set of directions

with suitable features are used to generate a ﬁnite set of points

at which the objective function is evaluated. An example of such

a set of directions is a positive basis. A ﬁnite set, or and inﬁnite

set of positive bases maybe used during the algorithm. Another

example is an integer lattice, which is constructed from a positive

basis. A well known class of mesh based directional direct search

methods is Mesh Adaptive Direct Search (MADS), proposed by Au-

det and Dennis in 2006 [103]. The general structure of a DDS meth-

od is as follows [104]:

procedure DirectSearch

begin

Initialize x

0

and a set of directions D;

while (not termination condition) do

begin

Search for a point with f(x)<f(x

k

) (optional);

Poll points from fx

k

þ

a

k

d:d2D

k

ð2 DÞg;

if f(x

k

+

a

k

d

k

)<f(x

k

)

Stop polling;

x

k+1

x

k

+

a

k

d

k

;

else

x

k+1

x

k

;

Update mesh parameter

a

k

;

end

end

end

DDS methods have been applied to structural engineering prob-

lems such as optimizing braced steel frameworks [105], structural

damage detection [106], and design optimization of reinforced

concrete ﬂat slab buildings [33] and viscous dampers [35].In

[82], a set of current conﬁgurations are used with a simulated

annealing framework to create a direct search simulated annealing

(DSA) method for design optimization of laminated composite

structures.

5.2. Simplicial direct search

In Simplicial Direct Search (SDS) methods, the algorithm evalu-

ates the function at a set of points that form a simplex and uses

those function values to decide the next move. A simplex in R

n

is

the convex hull of a set of n+ 1 afﬁnely independent points. By

evaluating the function at a set of points that forms a simplex,

the algorithm collects sufﬁcient information from around the cur-

rent iterate. (A shifted set of n+ 1 afﬁnely independent points

forms a set of linearly independent points, i.e., the shifted set spans

24 W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28

Author's personal copy

R

n

.) The most well known simplex based simplicial direct search

method is the Nelder-Mead method [107] (also known as the

NM, the amoeba or the adaptive simplex method). We note that

the original Nelder-Mead method proposed by Nelder and Mead

in 1965 [107] does not have convergence theory, but many vari-

ants of the method do.

SDS methods have been used in several structural engineering

applications, including structural damage identiﬁcation [86], truss

design optimization [27] and estimation of a crack location and

depth in a cantilever beam [37]. See also [36].

5.3. Simplex gradient methods

A simplex gradient method (SGM) uses a simplex gradient in-

stead of the true gradient to generate search directions that point

towards nearby (local) minimizers. A simplex gradient is the gradi-

ent of the linear interpolation over a set of n+ 1 points in R

n

. Unlike

SDS methods, which use simplices to provide a set of directions to

evaluate the function on, SGMs calculate simplex gradients to ﬁnd

descent directions. The general structure of an SGM method is as

follows [104]:

procedure SimplexGradient

begin

Initialize x

0

, simplex Y

0

, search radius

D

0

and simplex

accuracy measure

l

k

;

Initilize line search Armijo-like parameter

g

;

while (not termination condition) do

begin

Compute a simplex gradient r

S

f(x

k

) such that

D

k

6

l

k

kr

S

f(x

k

)k;

Line search: Find t

k

> 0 such that f(x

k

t

k

)

r

S

f(Y

k

)) < f(x

k

)

g

t

k

kr

S

f(Y

k

)k

2

;

if do not ﬁnd t

k

>0

Decrease

l

k

;

else

Let x

kþ1

¼arg min

y2S

k

ffðyÞg, where S

k

contains all

f-evals from this iteration;

end

end

end

An application of an SGM in structural engineering is seen in

[55] for welded beam and coil spring design.

The Robust Approximate Gradient Sampling (RAGS) algorithm is

a novel derivative-free optimization algorithm for ﬁnite minimax

problems, proposed by Hare and Nutini in 2012 [108].TheRAGS

method is an improvement on SGMs for structured functions. By

exploiting the substructure of the ﬁnite max function, the RAGS algo-

rithm is able to minimize along non-differentiable ridges of non-

smooth functions and converge to minima of the objective function.

The general structure of the RAGS algorithm is as follows [108]:

procedure RAGS

begin

Initialize x

0

, search radius

D

0

, Armijo-like parameter

g

and

other parameters;

begin

Generate a set of n+ 1 points;

Use points to generate robust approximate subdifferential G

k

Y

;

Set search direction d

k

Y

¼Proj 0jG

k

Y

;

if

D

k

small, but jd

k

jlarge

Carry out line search: ﬁnd t

k

> 0 such that

f(x

k

+t

k

d

k

)<f(x

k

)

g

t

k

jd

k

j

2

;

Success: update x

k

and loop;

Failure: decrease accuracy measure and loop;

else if

D

k

large

Decrease

D

k

and loop;

else

Terminate;

end

end

end

In [38], the RAGS algorithm is shown to be a quick converging

and efﬁcient method for solving the problem of minimizing the

maximum inter story drift between two buildings.

5.4. Trust-region methods

Trust-region (TR) methods locally minimize quadratic models

of the objective function over regions where the quadratic model

is ‘‘trusted’’ to be accurate. TR methods are Derivative-free Optimi-

zation methods, with an abundant number of publications illus-

trating the supporting convergence analysis and theory. An

example of a TR method used in a practical application can be

found in [109], where the design of a vehicle door is optimized.

To the authors’ knowledge, TR methods have not been applied to

any applications in structural engineering to date.

5.5. Strengths and limitations of derivative-free optimization

Derivative-free Optimization’s strongest aspect is its mathe-

matical convergence theory that guarantees the quality of the ﬁnal

solution. This makes Derivative-free Optimization very well suited

for applying as a ﬁnal step to ensure local optimality. However,

Derivative-free Optimization methods typically scale poorly with

dimension, so many require very large numbers of function calls

in problems where the number of variables is very large.

6. Discussion and conclusion

In the previous sections, we provided multiple references that

use non-gradient methods in structural engineering applications.

We provide a summary of the methods in Table 2.InTable 1 below,

we summarize a few of the papers that compared the performance

of several non-gradient methods on one application. While many

other papers compared various non-gradient methods, we limit

ourselves to those that do the comparison using a structural engi-

neering problem and use at least two algorithms discussed in this

article.

1. Other memetic methods.

2. Directional direct search (MADS) and URAGS.

3. UEvolutionary strategies.

4. UAdaptive harmony search.

5. Branch-and-bound metaheuristic.

6. UBee colony optimization and simplex method.

It is obvious that non-gradient methods are well used in struc-

tural engineering applications. As seen in Table 1, there are three

papers that compare non-gradient methods with gradient based

methods [43,54,55].In[43], the presented PSO method is shown

to generate comparable results to several gradient based methods.

W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28 25

Author's personal copy

In [55], the presented bee colony algorithm is also shown to gener-

ate comparable results to a gradient based method. In [54], the gra-

dient based methods win. As stated before, non-gradient methods

are useful when gradient information is unavailable, unreliable or

expensive in terms of computation time. However, when com-

pared against a gradient based method for a function with gradient

information, a non-gradient method will almost always come up

short.

We also observe from Table 1, as well as Section 2, that evolu-

tionary algorithms are the most commonly used non-gradient

methods in structural optimization. Both discrete and continuous

problems can be handled by evolutionary algorithms, as well as

constrained or unconstrained problems. Indeed, GAs are very ver-

satile with respect to the types of problems they can be applied

to. (For a complete summary table of the methods presented in this

paper and the types of problems they can be applied to, see Table 2

at the end of this section.)

However, this does not necessarily imply that evolutionary

algorithms are the most appropriate method for black-box prob-

lems in structural engineering. In fact, we see multiple papers

using evolutionary algorithms as benchmark methods to compare

other methods against (see the end of Section 2.1). In all of these

papers, evolutionary algorithms are shown to perform comparably

or worse with respect to efﬁciency and solution quality. This obser-

vation does spark the suggestion that evolutionary algorithms may

be overused, speciﬁcally for continuous problems. As evolutionary

algorithms were originally designed for discrete problems, it is not

surprising to see an evolutionary algorithm come in second place

to an algorithm designed for continuous problems.

This being said, it is worth noting that it would be inaccurate to

conclude that evolutionary algorithms are ‘bad’. Most of the papers

in question focus on newly designed algorithms by the papers’

authors. As such, evolutionary algorithms used may not have been

optimally adjusted to the problem in question. It is safe to say that

it may be beneﬁcial to use a method designed to deal with the spe-

ciﬁc structure of the problem under consideration.

We see methods designed for speciﬁc problems in the area of

Derivative-free Optimization. Since these algorithms have support-

ing convergence theory, speciﬁc assumptions are usually made

about the objective function. Supporting convergence theory al-

lows us to escape the uncertainty of a heuristic method; we know

that when a Derivative-free Optimization algorithm terminates, it

has found a locally optimal (or in the case of a convex function,

globally optimal) solution. Furthermore, Derivative-free Optimiza-

Table 2

Summary of Methods: section number, algorithm, abbreviation for algorithm, description of algorithm, if algorithm is used for constrained or unconstrained, discrete or

continuous optimization problems (bold indicates algorithm is primarily used for problems of that type) and any additional assumptions on the problem.

Section Algorithm Abbr. Description Constrained/ Discrete/ Other assumptions

unconstrained continuous on problem

2.1 Genetic GA Evolutionary gene- constrained and discrete and –

algorithm based heuristic unconstrained continuous

3.1 Harmony HS Music inspired Constrained and Discrete and –

search heuristic unconstrained continuous

3.2 Simulated SA Materials science Constrained and Discrete and Large

annealing based heuristic unconstrained continuous search space

3.3 Ray RO Light ray Constrained Continuous –

optimization based heuristic

3.4 Tabu TS Local search Constrained Discrete and Combinatorial

search heuristic continuous

4.1 Ant colony ACO Stochastic pheromone Unconstrained and Discrete Stochastic

optimization mimicking heuristic constrained combinatorial

4.2 Particle PSO Flocking behavior Unconstrained and Continuous and –

swarm opt. based heuristic constrained discrete

4.3 Shufﬂed SFL Evolutionary meme- Constrained and discrete Combinatorial

frog-leaping based heuristic unconstrained

4.4 Bee colony BCO Bee foraging Unconstrained and Continuous and Functional,

optimization based heuristic constrained discrete combinatorial

5.1 Directional DDS DFO mesh/lattice Unconstrained and Continuous Non-linear

direct search based algorithm constrained

5.2 Simplicial SDS DFO simplex Unconstrained and Continuous –

direct search based algorithm constrained

5.3 Robust approx. RAGS DFO substructure Unconstrained Continuous Finite minimax

grad. sampling exploiting algorithm problem

Table 1

Comparison of methods: reference, application description and algorithms compared (indicates that the corresponding algorithm was compared, Uindicates the ‘winning’

algorithm(s)).

Refs. Application Grad. GA HS TS SA ACO PSO SFL Other

[20] Bridge deck rehabilitation U1

[38] Seismic dampers 2

[40] Steel frames U

[30] Steel frames 3

[45] Steel frames 4

[83] Steel frames UU

[43] Truss-structures UU

[26] Truss-strucutres U5

[54] Truss-structures U

[42] Water distribution network design U

[55] Welded beam/coil design U6

26 W. Hare et al. / Advances in Engineering Software 59 (2013) 19–28

Author's personal copy

tion methods can easily incorporate a heuristic on top of their reg-

ular structure to decrease convergence time and increase solution

quality.

As seen in Table 1, heuristics such as SA and TS are commonly

used in structural engineering. Similar to evolutionary algorithms,

a heuristic has very few limitations as to what type of problem it

can be applied to. As stated above, heuristics are often used in con-

junction with other algorithms, which is just another way that

algorithms can be easily tailored to the problem at hand.

In conclusion, non-gradient methods are widely used in struc-

tural engineering applications. Most dominantly, we see heuristics

being applied to various problems. The strengths of these methods

include their ﬂexibility and versatility to be applied to multiple dif-

ferent problem types. For difﬁcult, restrictive problems, these

methods are easy to implement and can provide reasonable solu-

tions. However, by tailoring an optimization method to or using

a method that is tailored to the problem at hand, a signiﬁcant in-

crease in solution quality and efﬁciency of the algorithm can be

observed.

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