A modified probability collectives optimization algorithm based on trust region method and a new temperature annealing schedule

Abstract

This article presents a distributed random search optimization method, the trust region probability collectives (TRPC) method, for unconstrained optimization problems without closed forms. Through analyzing the framework of the original probability collectives (PC) algorithm, three potential requirements on solving the original PC model are first identified. Then an interior point trust region method for bound constrained minimization is adopted to satisfy these requirements. Besides, the temperature annealing schedule is also redesigned to improve the algorithmic performance. Since the new annealing schedule is linked to the gradient, it is much more flexible and efficient than the original one. Ten benchmark functions are used to test the modified algorithm. Numerical results show that TRPC is superior to the PC algorithm in iteration times, accuracy, and robustness.

Appendices

Appendix A

This appendix provides more detailed explanations about the interior TR method we use in Sect. 3. However, the proof of the convergence is still omitted here. Readers who are interested in this could check (Coleman et al. 1996) for more information.

The method we consider is mainly for the problem with a smooth nonlinear objective function subject to bounds on variables:

$$\begin{aligned} \mathop {\min }\limits _{\text {x}\in \text {R}^\text {n}} f(x),l\le x\le u, \end{aligned}$$

(13)

The feasible region is ${F}\mathop {\,=\,}\limits ^{\text {def}}\,\{x:l\le x\le u\}$ and the strict interior is $\mathrm{inf}({F})\mathop {\,=\,}\limits ^{\text {def}} \{x:l<x<u\}$. It can be easily checked that model (10) follows this form with $l=0$ and $u=1$.

The main idea of this interior TR method is to transform the problem above into a corresponding TR sub-problem which has the same form with the TR sub-problem of an unconstrained problem. To implement this, first we define a vector $v(x)$ and an affine scaling matrix $D(x)$ as follows:

Definition 3

Let $v(x_{i})= (v_{i1}, v_{i2}, \ldots , v_{in})$ be a vector for $x_{i}$; then

$$\begin{aligned} v_i =\left\{ {\begin{array}{l@{\quad }l} x_i -u_i ,&{}\text {if}\;\nabla f(x)_i <0 \\ x_i -l_i ,&{}\text {if}\;\nabla f(x)_i \ge 0 \\ \end{array}} \right. \end{aligned}$$

Note that $v_{i}$ measures each component’s distance from the current point $x=(x_{1}, x_{2}, \ldots , x_{n})$ to the bound $l$ and $u$.

Definition 4

For all $v(x_{i})$, let

$$\begin{aligned} D(x) = {\text {diag}}(\left| {v(x_{i} )} \right| ^{{ - 1/2}} ), \end{aligned}$$

where diag($\cdot$) denotes a diagonal matrix.

Assume $x$* is a local minimizer for (13), so the first-order necessary conditions for $x$* should be

$$\begin{aligned} \mathrm{first}\,\mathrm{order}:\left\{ {\begin{array}{l@{\quad }l} \nabla f(x^*)_i =0, &{} \mathrm{if}\,l_i <(x^*)_i <u_i , \\ \nabla f(x^*)_i \le 0, &{} \mathrm{if}\,(x^*)_i =u_i , \\ \nabla f(x^*)_i \ge 0, &{}\mathrm{if}\,(x^*)_i =l_i , \\ \end{array}} \right. \end{aligned}$$

It is worth noting that these first-order conditions to (13) are equivalent to

$$\begin{aligned} D_*^{-2} \nabla f(x^*)=0 \end{aligned}$$

(14)

(14) has the form of the first-order conditions for unconstrained problems and this is exactly why we use the scaling transformation.

So, a Newton stop for (14) satisfies

$$\begin{aligned} (D_k^{-2} H_k +\mathrm{diag}(\nabla f(x_k ))J_k^v )d_k =-D_k^{-2} \nabla f(x_k ), \end{aligned}$$

where $J_k^v$ is the Jacobian matrix of $\vert v(x_{i})\vert$; we set $J_{k}=\mathrm{diag}$(sign($\nabla f))$. The term $\mathrm{diag}(\nabla f(x_k ))J_k^v$ on the left side is to make the scaled Hessian matrix $D_k^{-2} H_k$ positive semi-definite.

Based on this Newton step, we could define our quadratic model for the TR method:

$$\begin{aligned} \begin{array}{l} \psi _k (s)=s^T\nabla f_k +\frac{1}{2}s^TM_k s \\ \mathrm{s.t.}\,\left\| s \right\| \le \delta _k, \\ \end{array} \end{aligned}$$

(15)

where

$$\begin{aligned} C(x)=D(x)\mathrm{diag}(\nabla f(x))J^v(x)D(x), \end{aligned}$$

$$\begin{aligned} M(x)=B(x)+C(x). \end{aligned}$$

And a slight modification of the quadratic model above is exactly (11) we used in Sect. 3.2 where $\nabla f_k =\nabla L_k$. It’s also obvious that the statements a$\sim$c in Sect. 3.2 hold.

Appendix B

This appendix describes the dog-leg method we used to solve the quadratic sub-problem in TR. Still, we omit the convergence proof of this method and only present the results. More information could be found in Gill et al. (1981).

The quadratic sub-problem we consider is

$$\begin{aligned} \begin{array}{l} \min m_c (x_c +s)=f(x_c )+\nabla f(x_c )^Ts+\frac{1}{2}s^TH_c s, \\ \mathrm{subject}\,\mathrm{to}\,\left\| s \right\| _2 \le \delta _c . \\ \end{array} \end{aligned}$$

(16)

Note if we define $\psi (s)=m_c (x_c +s)-f(x_c )$, (16) has the same form of (15).

Thedog-leg method does not find the optimal solution $s$* to (16). Instead, it uses two directions to approximate $s$*. The first direction is the steepest descent direction and the second is the Newton direction. According to the trust region radius, different $s$ is chosen as an approximate solution to (16). This idea could be illustrated by the figure below (Fig. 16).

Fig. 16

TR method

Point C.P. is the Cauchy Point, the minimizer of (16) in the steepest descent direction. Obviously, we need to consider three cases:

Case 1 if $\delta _c \le \left\| {s^\mathrm{C.P.}} \right\| _2$

In this case, we choose $s=\alpha s^\mathrm{C.P.},\,0<\alpha <1$ as the final solution.

Case 2 if $\left\| {s^\mathrm{C.P.}} \right\| _2 \le \delta _c \le \left\| {s^N} \right\| _2$

In this case, we choose $s=s^\mathrm{C.P.}+\alpha (s^N-s^\mathrm{C.P.}),\,0<\alpha <1$ as the final solution.

Case 3 if $\delta _c \ge \left\| {s^N} \right\| _2$

In this case, we choose $s=s^N$ as the final solution

It has been proved that alone the curve $x_{c}\rightarrow$C.P. $\rightarrow x_{N}$, $m_{c}$ decreases monotonically and there is always $\left\| {s^\mathrm{C.P.}} \right\| _2 \le \left\| {s^N} \right\| _2$. So every solution generated by the dog-leg method to the quadratic sub-problem is a sufficient descent and the method ultimately converges. In Sect. 3.2, we use the notation $s^{B}$ and $s^{U}$ instead of $s^{N}$ and $s^\mathrm{C.P.}.$