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.
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This work was supported in part by the National Science Foundation under grant CMMI 08-25677 and in part by the ONR (HUNT) grant N00014-08-1-0696 and the AFOSR (MURI) grant FA9550-06-1-0312.
Appendix A
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
Now note that the following is true:
because
and
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
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:
Then combining (38) and (39), f(λ) is defined for all λ∈ℝ. When λ≠0 we compute the derivative of f(λ) as
Next notice that
with
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
which implies
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
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
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
From (46) and (47) it follows that F(λ) is nondecreasing when λ>0 and is nonincreasing when λ<0. Thus,
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
where
Finally,
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
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
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
In addition, note that
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+. □
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Stipanović, D.M., Tomlin, C.J. & Leitmann, G. Monotone Approximations of Minimum and Maximum Functions and Multi-objective Problems. Appl Math Optim 66, 455–473 (2012). https://doi.org/10.1007/s00245-012-9179-8
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DOI: https://doi.org/10.1007/s00245-012-9179-8