Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems

Abstract

In this paper the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are derived using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example with two evaders and two pursuers is provided.

Appendix A

In the Appendix we provide proofs for Theorems 1 and 2.

1.1 A.1 Proof of Theorem 1

First, the values of the minimum approximation functions (1) and (2) can be written as

$$ \everymath{\displaystyle} \begin{array}{lll} \underline{\sigma}(\delta,a) &=& \frac{a_m}{\sqrt[\delta]{1+\sum_{i \ne m} (a_m/a_i)^\delta}},\\[6pt] \overline{\sigma}(\delta,a) &=& \frac{a_m \sqrt[\delta]{N}}{\sqrt[\delta]{1+\sum_{i\ne m}(a_m/a_i)^\delta}}. \end{array} $$

(36)

Now note that the following is true:

$$ \lim_{\delta \to +\infty } \sqrt[\delta]{1+\sum_{i=1}^N c_i^\delta}=1\quad \hbox{if}\ \bigl(\forall i \in \mathbf{N}\bigr) \bigl(c_i \in [0,1]\bigr) $$

(37)

because

$$1\le \sqrt[\delta]{1+\sum_{i=1}^N c_i^\delta} \le \sqrt[\delta]{1+N}\quad \hbox{if}\ (\forall i\in \mathbf{N}) \bigl(c_i \in [0,1]\bigr), $$

and

$$\lim_{\delta \to +\infty} \sqrt[\delta]{1+N}=1\quad \hbox{for any positive integer}\ N. $$

Since a _m /a _i ≤1 for all i∈N, N≥1+∑ _i≠m (a _m /a _i )^δ, and by using (37), we conclude that (6) and inequalities $\underline{\sigma}_{a}(\delta) < a_{m} \le \overline{\sigma}_{a}(\delta)$ for any finite δ>0 are valid. In addition, the minimum approximation functions behave well for any finite positive δ when the minimum approaches zero, that is, (7) which is a direct consequence of (36).

To complete the proof we need to show that $\underline{\sigma}_{a}(\delta)$ increases with δ and that $\overline{\sigma}_{a}(\delta)$ does not increase with δ. In order to do so, we use some results on inequalities related to means provided in [25] and also in [6, 7]. It is important to note that some historical facts related to derivations of results involving inequalities for means can be found on p. 77 in [25]. Therefore, we recall the function in [25] defined by

$$ f(\lambda)= \Biggl(\sum_{i=1}^N k_i a_i^\lambda \Biggr)^{1/\lambda} $$

(38)

where all parameters k _i and a _i for all i∈N are given positive numbers and λ∈ℝ∖{0}. Now, let us define f(0)=lim _λ→0 f(λ) and using Bernoulli’s result on computing limits (more commonly known as L’Hospital’s rule), we obtain:

(39)

Then combining (38) and (39), f(λ) is defined for all λ∈ℝ. When λ≠0 we compute the derivative of f(λ) as

(40)

Next notice that

$$ \underline{\sigma}_a(\delta)=f_1(-\delta),\quad \delta >0 $$

(41)

with

$$ f_1(\cdot)\equiv f(\cdot)|_{k_1=k_2=\ldots=k_N=1} $$

(42)

where ≡ and (⋅) are used to denote that the two functions are equal, and $f(\cdot)|_{k_{1}=k_{2}=\cdots=k_{N}=1}$ denotes function f(⋅) when the parameter values are set to k ₁=k ₂=⋯=k _N =1. Now we proceed by computing

(43)

which implies

$$ \frac{d \underline{\sigma}_a(\delta)}{d \delta} = \frac{f_1(-\delta)}{\delta^2(\sum_{i=1}^N a_i^{-\delta})} \Biggl(\sum_{i=1}^N a_i^{-\delta} \Biggl(\ln\Biggl(\sum_{j=1}^N a_j^{-\delta}\Biggr) - \ln\bigl(a_i^{-\delta}\bigr) \Biggr) \Biggr). $$

(44)

Since ln(y) increases with y∈ℝ₊, we conclude that $d \underline{\sigma}_{a}(\delta)/d \delta>0$, that is, $\underline{\sigma}_{a}(\delta)$ is strictly increasing with δ∈ℝ₊.

At this point, let us compute λ ² f′(λ)/f(λ) [25] for λ≠0 and denote this expression by F(λ) which is well defined for all λ∈ℝ since

$$ F(\lambda) =\lambda \frac{\sum_{i=1}^N k_i a_i^\lambda \ln(a_i)}{\sum_{i=1}^N k_i a_i^\lambda} - \ln \Biggl(\sum_{i=1}^N k_i a_i^\lambda\Biggr). $$

(45)

Notice that, since f(λ)>0, F(λ) and f′(λ) have the same sign when λ≠0. Now, F′(λ)=dF(λ)/dλ can be computed as

(46)

Let $d_{i}(\lambda)=k_{i} a_{i}^{\lambda}$, e _i =ln(a _i ), and $F'(\lambda)=\lambda A(\lambda) (\sum_{i=1}^{N} k_{i} a_{i}^{\lambda})^{-2}$ where

(47)

From (46) and (47) it follows that F(λ) is nondecreasing when λ>0 and is nonincreasing when λ<0. Thus,

$$ F(0)=-\ln \Biggl(\sum_{i=1}^N k_i \Biggr) $$

(48)

is the minimum of F(λ), λ∈ℝ. Then, related to the $\overline{\sigma}_{a}(\delta)$ case, that is, when k _i =1/N implies $\sum_{i=1}^{N} k_{i} =1$, we obtain that F(0)=0. Therefore, F(λ)≥0 when λ≠0 and F(0)=0. This implies f′(λ)=f(λ)F(λ)/λ ²≥0 for λ≠0. In addition notice that

$$ \overline{\sigma}_a(\delta)=f_2(-\delta),\quad \delta >0 $$

(49)

where

$$ f_2(\cdot)\equiv f(\cdot)|_{k_1=k_2=\cdots=k_N=1/N}. $$

(50)

Finally,

$$ \frac{d \overline{\sigma}_a(\delta)}{d \delta} = \frac{df_2(\lambda)}{d\lambda} \frac{d\lambda}{d \delta} = -\frac{d f_2(\lambda)}{d\lambda} $$

(51)

implies that $\overline{\sigma}_{a}(\delta)$ is nonincreasing for δ∈ℝ₊.  □

1.2 A.2 Proof of Theorem 2

First note that the values of the approximation functions (8) and (9) can be rewritten as

$$ \everymath{\displaystyle} \begin{array}{lll} \underline{\rho} (\delta,a) &=& a_M{\frac{\sqrt[\delta]{1+\sum_{i \ne M} (a_i/a_M)^\delta}}{\sqrt[\delta]{N}}}, \\[12pt] \overline{\rho} (\delta,a) &=& a_M \sqrt[\delta]{1+\sum_{i\ne M}(a_i/a_M)^\delta}. \end{array} $$

(52)

Using (37), the fact that a _i /a _M ≤1 for all i∈N, and N≥1+∑ _i≠M (a _i /a _M )^δ, we conclude that (13) and $\underline{\rho}_{a} (\delta) \le a_{M} < \overline{\rho}_{a} (\delta)$, δ∈ℝ₊, are both valid.

Now, note that $\underline{\rho}_{a}(\cdot) \equiv f(\cdot)|_{k_{1}=\cdots=k_{N}=1/N}$ implies

$$ \frac{d \underline{\rho}_a(\delta)}{d \delta} = \frac{d f (\delta)}{d\delta}\bigg|_{k_1=\cdots=k_N=1/N} $$

(53)

and therefore $d \underline{\rho}_{a} (\delta)/d \delta >0$ so that $\underline{\rho}_{a} (\delta)$ is nondecreasing for δ∈ℝ₊. Also notice that $\overline{\rho}_{a}(\cdot) \equiv f(\cdot)|_{k_{1}=\cdots=k_{N}=1}$ which implies

$$ \frac{d \overline{\rho}_a(\delta)}{d \delta} = \frac{d f (\delta)}{d\delta}\bigg|_{k_1=\cdots=k_N=1}. $$

(54)

In addition, note that

$$ \sum_{i=1}^N a_i^\lambda \ln \bigl(a_i^\lambda\bigr) - \sum_{i=1}^N a_i^\lambda \ln \Biggl(\sum_{j=1}^N a_j^\lambda\Biggr)= \sum_{i=1}^N a_i^\lambda \ln \biggl( \frac{a_i^\lambda}{\sum_{j=1}^N a_j^\lambda} \biggr) <0 $$

(55)

for any $a=[a_{1},\ldots,a_{N}]^{T} \in \mathbb{R}^{N}_{+}$. This implies that $d \overline{\rho}_{a} (\delta) /d \delta <0$ so that $\overline{\rho}_{a} (\delta)$ is decreasing for δ∈ℝ₊.  □