ON THE MIXED INTEGER RANDOMIZED PATTERN SEARCH ALGORITHM

We analyze the convergence and performance of a novel direct search algorithm for identifying at least a local minimum of unconstrained mixed integer nonlinear optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a randomized pattern search, which is modified by two main operations for finding at least a local minimum of our problem, namely: moving operation and shrinking operation. The convergence properties of the MIRPSA are here analyzed from a Markov chain viewpoint, which is represented by an infinite countable set of states {d(q)}∞ q=0 , where each state d(q) is defined by a measure of the qth randomized pattern search Hq, for all q ∈ N. According to the algorithm, when a moving operation is carried out on a qth randomized pattern search Hq, the MIRPSA Markov chain holds its state. Meanwhile, if the MIRPSA carries out a shrinking operation on a qth randomized pattern search Hq, the algorithm will then visit the next (q + 1)th state. Since the MIRPSA Markov chain never goes back to any visited state, we therefore say that the MIRPSA yields a birth and miscarriage Markov chain.


Introduction
Consider the following unconstrained mixed integer nonlinear problem: minimize (x,y)∈R n ×Z m f (x, y), (1.1) where f (x, y) : R n × Z m → R is a nonlinear objective function, for which an analytical and explicit mathematical expression cannot be obtained. In this case, the objective function must be evaluated by an appropriate simulation model, and x ∈ R n and y ∈ Z m are decision variables of our problem. Of course, there exists no derivative function due to the objective function being defined in the set R n × Z m . This class of problems often arises out of practical applications, which are frequently studied in the engineering design field, wherein there exist both real and discrete variables. Examples of these problems are shown by Floudas [3], who presents a wide set of chemical engineering problems. Moreover, mixed integer 486 EBERT BREA optimization problems are also applied in the design of electrical motors. For example, Brea [2] presents an application of this problem for optimal design of a permanent magnet electric machine.
For identifying at least a local minimum of the problem given by Eq. (1.1), we have considered the use of randomized pattern search in the Euclidean space R n × Z m . The idea of using a randomized pattern search was put forward by Lawrence and Fawzi [8,7], who proposed an algorithmic method of linear and nonlinear optimization problems when the set of decision variables are real. Nevertheless, the authors do not analyze any convergence property of their algorithm.
There exists an abundant literature about the use of deterministic pattern search for identifying optimum solution in nonlinear optimization problems; for instance, Kolda and coworkers [5,6], and also Torczon [11] do a deep study on convergence of pattern search algorithms. In addition, Hart [4] presents a convergence analysis of evolutionary pattern search, wherein he analyzes a class of evolutionary algorithm, based on pattern search algorithm, for unconstrained and bound constrained optimization problems. Audet and Dennis [1] also propose a pattern search algorithm for mixed integer optimization problems.
Nevertheless, the author here proposes a novel randomized pattern search for finding solutions to mixed integer unconstrained optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a mixed integer randomized pattern search, which is modified by the MIRPSA for identifying at least a local minimum of our nonlinear optimization problems.
The MIRPSA is easy to code; however, studying the properties of its convergence is not so straightforward. Moreover, this can be considered a great challenge. We study its convergence properties using a Markov chain approach, where each state d(q) for all q ∈ N represents a measure of the randomized pattern search H k at each kth iteration of the algorithm. Therefore, the Markov chain is an infinite countable set of state space Λ = {d(q)} ∞ q=0 . Our goal is focused on convergence properties of the MIRPSA using a Markov chain viewpoint, because of the fact that each qth state is given by a measure of the randomized pattern search, and allows us to estimate the probability of being at each qth state.
The MIRPSA starts with an initial guess 0th mixed integer randomized pattern search H 0 , which is used for locating an initial set of randomized trial points, in order to evaluate the objective function at each trial point for comparing with the center of the randomized pattern search function value f (c k ). Depending on these comparisons with the center of the randomized pattern search function value f (c k ), can be executed a moving operation or shrinking operation on the current mixed integer randomized pattern search H 0 . Namely the MIRPSA is basically based on two main operations: moving operation and shrinking operation, on any kth mixed integer randomized pattern search H k , which is used for locating a set of randomized trial points at each kth iteration around the center of the randomized pattern search c k . The pattern search center will be moved to the best trial point, if the objective function value at the best trial point is better than the objective function value at the current center c k . In contrast, the pattern search center will remain unchanged, but pattern search range will shrink for both real and integer factors, if the objective function value at the best trial point does not improve the objective function at the center of the current pattern search. This procedure is iteratively repeated until some stopping rule is satisfied.
The remainder of the paper is arranged as follows. A general description, together with a pseudocode of the MIRPSA, are provided in Section 2. In Section 3 we describe a general Markov chain framework that is produced when the MIRPSA is being carried out in order to analyze its convergence properties. A low dimension case is studied in Section 4 and a simple numerical example is also reported. In Section 5 we show the performance of the MIRPSA when the objective function is a multidimensional quadratic function, and we also report results of some numerical test problems. Finally, conclusions and suggestions for future research are provided in Section 6.

The mixed integer randomized pattern search algorithm
The MIRPSA, displayed in Fig. 1, is based on a randomized pattern search H k , which is defined by a set of random trial points at each kth iteration for identifying at least a local optimum point of the problem presented by Eq. (1.1). Each random trial point z ∈ R n × Z m is generated by random locations around each kth pattern search center c k = (c (1) , . . . , c (n) , c (n+1) , . . . , c (n+m) ) t ∈ R n × Z m using continuous and discrete uniform distributions. Each ith real component of each trial point is randomly generated by meanwhile their integer components are yielded by where δ [k] and ∆ [k] denote, respectively, the parameters of continuous and discrete uniform distribution at each pattern search H k . The MIRPSA has two main operations: moving operation and shrinking operation, of which only one can be carried out on the random pattern search at each kth iteration. A moving operation is executed at the kth iteration, when a trial point z at least improves the current value objective function f (v k ). That is, if f (z) < f (v k ), then the center of the current randomized pattern search will be moved to the best trial point of the set of M random trial points yielded by the MIRPSA at each kth iteration. Meanwhile a shrinking operation will be carried out at the kth iteration, if the current pattern search center is better than all trial points at the current kth iteration. In this last case, the size of the randomized pattern search is shrunk by the algorithm at a rate of α in order to reduce the range of a continuous uniform distribution, and β for shrinking the step-length of a discrete uniform distribution.

On the convergence of the MIRPSA
In this section, we analyze the convergence of the MIRPSA from a Markov chain viewpoint, wherein the set of discrete state space is a countable and infinite Markov chain {d(q)} ∞ q=0 , where d(q) is given by a size measure of each kth random pattern search H k at each kth iteration of the MIRPSA for each qth state. The main convergence property is based on the fact that the MIRPSA visits each qth state of the Markov chain {d(q)} ∞ q=0 , when the MIRPSA is applied to a convex minimization problem.
Consider the MIRPSA as a Markov chain, where the set of state space is defined by {d(q)} ∞ q=0 . Here d(q), for each q ∈ N, represents a measure of the pattern search size of H k at each kth iteration, which holds on a qth state within a measurable set of N q successive moving operations that have been carried out at each kth iteration.
Thus, for each k ∈ N, d(k) = δ [k] + ∆ [k] , and this is computed by where N i is the amount of successive moving operations that are carried out by the MIRPSA at each ith state. Fig. 2 depicts the Markov chain of the MIRPSA, where, without loss of generality, each d(q) state is denoted by its qth number in order to simplify the notation in the figure. We can say that this Markov chain represents a birth and miscarriage discrete process, because at any kth iteration, the process may go from each qth state to the next (q + 1)th state, so called birth, or it may remain at its qth state, what would be considered a miscarriage. Notice that the MIRPSA never goes back from any qth state to any visited state {d(q)} q−1 q=0 . The MIRPSA holds on the same qth state due to a set of k successive moving operations, for all where N i is the total number of iterations, wherein the MIRPSA remains at an ith state. Meanwhile, it changes from a qth state into the following (q + 1)th state because of a shrinking operation, which takes place at each ( We can assert that for all and q ∈ N, the probability that any moving operation has taken place is given by On the other hand, we also say that for each k = q i=0 N i and q ∈ N, the probability that a shrinking operation has been carried out at each kth iteration is We denote the initial distribution by where the subscript 0 of π 0 (d(0)) means the 0th iteration, and the argument represents the state d(0). We similarly say that at each kth iteration, the qth state distribution is denoted by 3.1. Estimating the probability of miscarriage. We will show that the MIRPSA obtains h trial points from M samples of the pattern search H q with binomial distribution.

Proposition 3.1. Assume that a measure of size of any kth mixed integer randomized pattern search
. If the stochastic process {d(q)} ∞ q=0 , whose state is given by where N i is the amount of successive moving operations carried out by the MIRPSA at each qth state, or d(k + 1) = αδ [k] + β∆ [k] for all k ∈ N, then the MIRPSA is a non homogeneous Markov chain.
Proof. The MIRPSA goes from d(q) to d(q + 1) due to a shrinking operation, wherein the parameters of both continuous and discrete uniform distributions are respectively updated by δ [q+1] = αδ [q] and ∆ [q+1] = β∆ [q] , what causes a new uniform distribution at the qth state. Finally, since only one shrinking operation is carried out by the algorithm at each qth state, and only sampling trial of the current mixed integer randomized pattern search H k ∈ R n × Z m are used to determine where to move next, {d(q)} ∞ q=0 has the memoryless property, what causes that the MIRPSA just goes from a d(q) to d(q + 1) for all q ∈ N, and this fact allows us to ensure that the stochastic process is a non homogeneous Markov chain.

Remark 3.2.
Note that each kth mixed integer randomized pattern search H k ∈ R n × Z m is contained in a convex hull C(H k ), and its (n + m) dimensional volume for each qth state is given by Proof. Since the MIRPSA updates at each qth state because a shrinking operation is carried out, then Of course, the previous proposition needs proving the assumption that each qth state is non absorbing, which will be proved later, in Lemma 3.6.
j ∈ R n × Z m be the jth trial point of the kth mixed integer randomized pattern search H k , and let ν q denote the maximum number of iteration wherein the MIRPSA is at the qth state. Assume M sampling of trial points of a kth randomized pattern search H k , which are independent Bernoulli trials with probability  Then the probability that a moving operation takes place at each (k + ν)th iteration of a qth state is given by Proof. Due to the fact that the MIRPSA takes M independent Bernoulli trial points on the qth randomized pattern search H q , the number h of all the jth trial points whosef (z Hence, from Proposition 3.4, we can also assert As we can see each qth center point c [q,k+ν] is known from the best trial point at the (k + ν − 1)th iteration for each ν.
Notice that for each (k + ν)th iteration and for each jth random trial point It is worthwhile pointing out that ρ q,k+ν = P{f(z k+ν] )}; this mainly depends on the dimension of the problem and pattern search size.
We now turn to the estimation of the probability of miscarriage at each d(q) state of the MIRPSA. Theorem 3.5. Let ν be the quantity of iterations that has been carried out by the MIRPSA at any qth state d(q), for denoting the qth state without affecting the generality. The probability that then occur ν iterations at the qth state of the MIRPSA (this means ν moving operations after k iteration of the MIRPSA), is given for each k ∈ N, ν ∈ N + , q ∈ N + by q,q is the probability that a moving operation occurs, that is, the probability of keeping the qth state at [·] iteration; p [·] q−1,q is the probability that a shrinking operation has taken place at [·] iteration.
We now consider the case k + 1, wherein for estimating P{D k+1 = d(q)} we have for each k ∈ N, q ∈ N: Then, from Eq. (3.7), we have which yields for each k ∈ N, q ∈ N + : Using Eqs. (3.1) and (3.2), we get for all k ∈ N, q ∈ N + : On the other hand, we know that for each k ∈ N, q ∈ N + : Substituting k by k + 1 in Eq. (3.10), we have what yields by replacing Eq. (3.10) into Eq. (3.11) for each k ∈ N, q ∈ N + : In general, we get for all k ∈ N, ν ∈ N + , q ∈ N + : where we have assumed that y x (·) = 1, for all x, y ∈ N such that x > y. The following lemma allows us to establish the fact that each state that belongs to {d(q)} ∞ q=0 is non absorbing, therefore the process of identifying a local minimum will never stop, if the stopping rule is not met. Proof. For proving the lemma, we must ensure that each qth visited state can be left. In other words, P N [q] +ν (d(q), d(q)) = 1 and P N [q] +ν (d(q), d(q + 1)) = 0, where N [q] given by N [q] = q i=0 N i means the number of iterations carried out by the algorithm for arriving to each qth state, and ν is the number of iterations performed by the algorithm within each qth state.
From Eq. (3.6), we then have that for M trial points +ν] )} > 0 for each iteration due to convexity of the objective function.
This fact allows us to ensure that P N [q] +ν (d(q), d(q)) given by Eq. (3.14) is different from 1 for all ν ∈ N, and also, we can assert that This last fact allows us to assure that the MIRPSA always goes from a qth state to the (q + 1)th state with probability ϕ (0 < ϕ < 1) until the stopping rule of the MIRPSA has been held. In our algorithm, the stopping rule is given by Proof. According to the algorithm, a moving operation at a kth iteration causes the updating of the pattern search center by replacing c k+1 by the best trial points in the current mixed integer randomized pattern search H k ∈ R n ×Z m , what allows us to assert that f (c k+1 ) < f (c k ).

Theorem 3.8. If the MIRPSA is applied to a convex objective function, then the MIRPSA converges to a local minimum at least.
Proof. Since the MIRPSA visits all qth non absorbing state d(q) in {d(q)} ∞ q=0 due to Lemma 3.6, and according to Proposition 3.4, the probability of a moving operation taking place is non zero, then an improvement of the objective function occurs due to Theorem 3.7. The MIRPSA therefore converges to a local minimum at least, when infinite states are visited.
At this stage we have proved that the MIRPSA allows us to identify a local minimum at least. However, we did not provide a theoretical study of the convergence rate of the MIRPSA, which will here be proposed by the next theorem and associated corollary. It is worthwhile to remark that the convex hull volume of each mixed integer randomized pattern search H k ∈ R n × Z m , at each qth state, allows us to prove the convergence rate of the MIRPSA. Theorem 3.9. Assume that there exist ∆ [0] > 1 and β ∈ R such that 0 < β < 1, and also suppose that the n + m dimensional volume v(q) of the convex hull C(H q ) at each qth state is given, according to Remark 3.2, by Eq. (3.4). Then, for any state counter q ∈ N, Eq. (3.16) can be rewritten as On the other hand, according to Remark 3.2 and taking into account Eq. (3.17), we have that the volume of each convex hull C(H q ) in the Euclidean space R n ×N m at each qth state is given by which obviously allows us to assure that

ln(β)
, and the volume v(q) of the convex hull C(H q ) therefore becomes smaller in volume at a rate of α n β m , for all 0 ≤ q <q − 1, or α n , for all q ≥q − 1. As a result of this, we can assert that the rate of shrinking is degenerated by increasing n and m.

A low dimension case
In what follows, we analyze a low dimension case, R×Z. This could be considered the simplest case; it is, however, a nontrivial case.

Estimating ρ q,k+ν
In order to simplify our notation and without loss of generality, we shall denote z [q,k] and c [q,k] without the superscript [q, k].
Note that the measure of the randomized pattern search H k is equal to H k+ν , because the MIRPSA is at the qth state for ν iterations.
Let W be a random variable given by a convex functionf (v) : R× Z → R, which is defined byf (2) ) t ∈ R 2×1 ; and the matrix A = a (11) a (12) a (21) a (22) , which is a positive definite matrix in R 2×2 , because we have assumed that our objective function is convex.
Note that the stationary point of the surface is given byv = (v (1) ,v (2) ) t = − 1 2 A −1 b, which does not necessarily belong to Euclidean space R × Z [10, Ch. 6]. As a consequence of these facts, we then have a random functionw of two random variables (x,ỹ) t ∈ R × Z given byw According to the MIRPSA, at each kth iteration, the objective function value at random pattern search center c = (c (1) , c (2) ) t is known as a result of either moving operation or shrinking operation at the (k − 1)th iteration, and its deterministic value is f (c). Note that at each kth iteration, the MIRPSA obtains a kth pattern search center, which will be denoted by c without the superscript [k] in order to simplify the notation. In addition, we have a set of M trial points z j and each jth random trial point with a value given byf (z j ).
At each kth iteration, we can have two main situations: the first one when the stationary pointv of the objective function is outside the kth random pattern search, and the second situation is when it is inside the kth pattern search H k .
x y c c (1) The center of the pattern is far away from the stationary pointv.
The center of the pattern is near the stationary pointv. Consider Fig. 3(a), which displays a kth pattern search of size 4δ [k] ∆ [k] in R × Z, with center at c = (c (1) , c (2) ) t . In addition, it shows the stationary point of the objective function surfacev = (v (1) ,v (2) ) t on the plane xy, which is not necessarily a minimum of our problem, and the center of the pattern is far away from the vertexv.
Meanwhile, a case when the center of the pattern is near the stationary point of the objective function surfacev is depicted in Fig. 3(b). The figure also shows on the x axis the points ω 1 (y) and ω 2 (y), whose difference lets us estimate the probability P{f (z j ) < f (c)} for the particular case (x,y) t .
It is worthwhile pointing out that both figures depict a set of level curves of the generic mixed integer convex quadratic objective function, because its integer variable y was converted to a real variable for graphically showing the objective function.

A numerical example. Letw be a random variable given bỹ
(4.5) This allows us to estimate the probability P{f (z j ) < f (c)} using Eq. (4.2), that is, the probability of carrying out a moving operation at any kth iteration, which is therefore given by where for this particular example and χ(y) is calculated using Eqs. (4.3) and (4.5).
According to the MIRPSA, at the kth iteration, the objective function value at random pattern search center c is known by both moving operation or shrinking operation carried out at the (k − 1)th iteration, and its deterministic value isf (c) = f [k] 0 . In addition, we have a set of M trial points z j and each jth random trial point with a value given byf (z j ).   All the initial points for the test problems of Appendix A were at 10 for each component, except for the test problem MRF, which was an initial point with each component fixed at 4.
The following tables report: mean (μ), standard deviation (σ) and range of both m(p, s) and distance to the true point (DTP) for a set of test problems, which are formulated in Appendix A.
As can be seen from Table 1, the MIRPSA had an adequate m(p, s) for the case of the test problem QF, because we get a mean value between 10 −7 and 10 −3 . Moreover, we obtained a small DTP mean, which yield values between 10 −3 and 4 for all dimensional cases, if taking into account the NE for each case.    Table 3, which allows us to assert that the MIRPSA achieved a good performance for all dimensional cases.
Finally, Table 4 displays the performance results of the test problem ADF, defined in Appendix A. As we can see from Table 4, the MIRPSA identified the optimum solution with 900 function evaluations, and a mean of m(p, s) = 1.570E-6.

Conclusions and future research
In this article, we have examined some convergence properties of the MIRPSA based on the approach of Markov chains. We have proved that the MIRPSA visits each qth state, and we have also proved that the MIRPSA has non absorbing states, which guarantees that the process of finding a local minimum stops when the stopping rule is satisfied, namely, only if α k δ [0] + β k ∆ [0] ≤ ε is satisfied, the MIRPSA will stop at kth iteration, and the MIRPSA therefore at least identifies a local minimum. Future research will focus on analyzing and tuning the MIRPSA using the approach of design of experiments. We also suggest to include some backtracking rules to the MIRPSA in order to study the performance of a modified MIRPSA for identifying global minimum of unconstrained nonlinear mixed integer problems.

Appendix A. Test problems
Here we present a set of mixed integer test problems, wherein we show the optimum solution of each test problem.
where a (i) ∈ R and b (i) ∈ R are all positive coefficients for all i = 1, . . . , n, and j = 1, . . . , m. In this paper a (i) and b (i) were all fixed at 1.  [9].