一、优化模型介绍
在所研究的区块链网络中，优化的变量为：挖矿决策（即 m）和资源分配（即 p 和 f），目标函数是使所有矿工的总利润最大化。问题可以表述为：
max m , p , f F miner = ∑ i ∈ N ′ F i miner s.t. C 1 : m i ∈ { 0 , 1 } , ∀ i ∈ N C 2 : p min ≤ p i ≤ p max , ∀ i ∈ N ′ C 3 : f min ≤ f i ≤ f max , ∀ i ∈ N ′ C 4 : ∑ i ∈ N ′ f i ≤ f total C 5 : F M S P ≥ 0 C 6 : T i t + T i m + T i o ≤ T i max , ∀ i ∈ N ′ \begin{aligned} \max _{\mathbf{m}, \mathbf{p}, \mathbf{f}} & F^{\text {miner }}=\sum_{i \in \mathcal{N}^{\prime}} F_{i}^{\text {miner }} \\ \text { s.t. } & C 1: m_{i} \in\{0,1\}, \forall i \in \mathcal{N} \\ & C 2: p^{\min } \leq p_{i} \leq p^{\max }, \forall i \in \mathcal{N}^{\prime} \\ & C 3: f^{\min } \leq f_{i} \leq f^{\max }, \forall i \in \mathcal{N}^{\prime} \\ & C 4: \sum_{i \in \mathcal{N}^{\prime}} f_{i} \leq f^{\text {total }} \\ & C 5: F^{M S P} \geq 0 \\ & C 6: T_{i}^{t}+T_{i}^{m}+T_{i}^{o} \leq T_{i}^{\max }, \forall i \in \mathcal{N}^{\prime} \end{aligned} m,p,fmax s.t. Fminer =i∈N′∑Fiminer C1:mi∈{ 0,1},∀i∈NC2:pmin≤pi≤pmax,∀i∈N′C3:fmin≤fi≤fmax,∀i∈N′C4:i∈N′∑fi≤ftotal C5:FMSP≥0C6:Tit+Tim+Tio≤Timax,∀i∈N′
其中：
C1表示每个矿工可以决定是否参与挖矿；
C2 指定分配给每个参与矿机的最小和最大传输功率；
C3 表示分配给每个参与矿工的最小和最大计算资源；
C4表示分配给参与矿机的总计算资源不能超过MEC服务器的总容量;
C5保证MSP的利润不小于0;
C6 规定卸载、挖掘和传播步骤的总时间不能超过最长时间约束。
在所研究的区块链网络中，我们假设 IoTD 是同质的，并且每个 IoTD 都具有相同的传输功率范围和相同的计算资源范围。
上式中：
F i m i n e r = ( w + α D i ) P i m ( 1 − P i o ) − c 1 E i t − c 2 f i , ∀ i ∈ N ′ R i = B log 2 ( 1 + p i H i σ 2 + ∑ j ∈ N ′ \ i m j p j H j ) , ∀ i ∈ N ′ T i t = D i R i , ∀ i ∈ N ′ T i m = D i X i f i , ∀ i ∈ N ′ E i m = k 1 f i 3 T i m , ∀ i ∈ N ′ P i m = k 2 T i m , ∀ i ∈ N ′ F M S P = ∑ i ∈ N ′ ( c 2 f i − c 3 E i m ) − c 3 E 0 P i o = 1 − e − λ ( T i o + T i s ) = 1 − e − λ ( z D i + T i t ) , ∀ i ∈ N ′ F_i^{miner}=(w+\alpha D_i)P_i^m(1-P_i^o)-c_1E_i^t-c_2f_i,\forall i\in\mathcal{N'}\\R_{i}=B \log _{2}\left(1+\frac{p_{i} H_{i}}{\sigma^{2}+\sum_{j \in \mathcal{N}^{\prime} \backslash i} m_{j} p_{j} H_{j}}\right), \forall i \in \mathcal{N}^{\prime}\\T_{i}^{t}=\frac{D_{i}}{R_{i}},\forall i\in\mathcal{N}^{\prime}\\T_{i}^{m}=\frac{D_{i}X_{i}}{f_{i}},\forall i\in\mathcal{N}'\\E_i^m=k_1f_i^3T_i^m,\forall i\in\mathcal{N}'\\P_i^m=\frac{k_2}{T_i^m},\forall i\in\mathcal{N}^{\prime}\\F^{MSP}=\sum_{i\in\mathcal{N}^{\prime}}\left(c_2f_i-c_3E_i^m\right)-c_3E_0\\\begin{aligned} P_{i}^{o}& =1-e^{-\lambda(T_{i}^{o}+T_{i}^{s})} \\ &=1-e^{-\lambda(zD_{i}+T_{i}^{t})},\forall i\in\mathcal{N}^{\prime} \end{aligned} Fiminer=(w+αDi)Pim(1−Pio)−c1Eit−c2fi,∀i∈N′Ri=Blog2(1+σ2+∑j∈N′\imjpjHjpiHi),∀i∈N′Tit=RiDi,∀i∈N′Tim=fiDiXi,∀i∈N′Eim=k1fi3Tim,∀i∈N′Pim=Timk2,∀i∈N′FMSP=i∈N′∑(c2fi−c3Eim)−c3E0Pio=1−e−λ(Tio+Tis)=1−e−λ(zDi+Tit),∀i∈N′
二、差分进化算法求解
2.1部分代码
close all
clear
clc
dbstop if all error
NP = 100;%矿工数量
para = parametersetting(NP);
para.MaxFEs =5000;%最大迭代次数
Result=Compute(NP,para);
figure(1)
plot(Result.FitCurve,'r-','linewidth',2)
xlabel('FEs')
ylabel('Token')
figure(2)
plot(Result.ConCurve,'g-','linewidth',2)
xlabel('FEs')
ylabel('Con')
2.2部分结果
当矿工数量为100时：所有矿工的利润随迭代次数的变化如下图所示
算法得到的资源分配：
1.99763301712028 0.222528597636855
1.98480090600989 0.232003797981878
1.99810737020089 0.516878075461127
1.99450954175327 0.121004799048830
1.98894335292950 0.457573161395314
1.98141441375851 0.764801153373885
1.99123792611056 0.0618336115864624
1.99957268156257 0.121004799048830
1.99869990696838 0.0545812896345451
1.99958167059988 0.555322442727203
1.99842776886770 0.0425674932800246
1.99782546212753 0.556999423219330
1.99781790486039 0.196587806899822
1.99507786088204 0.115226131066544
1.99052235611421 0.245674972808444
1.99670598640193 0.0505531222716088
1.99482731112569 0.570493296084591
1.99736278961552 0.483094177861634
1.98894335292950 0.262561711571175
1.98784689496156 0.0324778719744346
1.98851683245790 0.171964220456218
1.98796386190418 0.110054645825889
1.98418972990049 0.0724358226961023
1.99516235341290 0.0341179120870288
1.99873738363101 0.489382783726158
1.99697974388302 0.0173712437086769
1.98964833679332 0.0320026913839283
1.99751719786278 0.147890074497164
1.99751719786278 0.434936315273999
1.99748331769841 0.232003797981878
1.99960825876476 0.483665232586750
1.99763301712028 0.631745087572258
1.99703599779628 0.358292746434059
1.99528222092061 0.514944354258863
1.99655084169003 0.753834027257007
1.99842776886770 0.940560567187612
1.99836116767571 0.221230559879615
1.99981576341436 0.184249732087410
1.99836116767571 0.0324778719744346
1.99654201611710 0.335915952413277
1.99237903891650 0.155001423906853
1.99760611708088 0.375017552592607
1.99978704361437 0.561786832194378
1.98578574172372 0.0236239899979008
1.99866761178096 0.0324778719744346
1.99763301712028 0.472369465588862
1.99721838438050 0.700915679954801
1.99428564716577 0.157199586550231
1.99655135483398 0.105209390328771
1.94788362094720 0.0258755419701254
1.99449453062393 0.132251896484895
1.99700992290778 0.0898397719008559
1.99965518095321 0.596537124037070
1.99278786910748 0.0256042543513514
1.99957848431148 0.894961847587823
1.99175299365895 0.0890674637434230
1.99750797157559 0.607592532504797
1.99748331769841 0.0724358226961023
1.99260527116064 0.631745087572258
1.99928439965780 0.127930497832236
1.99817708666189 0.104282160660561
1.99421206141539 0.803656147079701
1.98359960108601 0.118868109287597
1.99899700099444 0.518357001275729
1.99528222092061 0.0324778719744346
1.99877098644022 0.665529673319171
1.99763301712028 0.334090268607101
1.99860560539076 0.0866379799536027
1.99979684848517 0.377299990245342
1.99855631180132 0.389679849807951
1.99731236573268 0.434936315273999
1.99696360320736 0.570493296084591
1.99993018378939 0.391296247028955
1.99965327995029 0.287460195344814
1.99979684848517 0.450997212108626
1.99751719786278 0.287460195344814
1.99763301712028 0.155001423906853
1.99783983352391 0.103569288167448
1.99654201611710 0.127930497832236
1.98747116264687 0.0330088002325308
1.99655135483398 0.0797018166113099
1.99108222250111 0.0866379799536027
1.99718273730151 0.662248213795699
1.99869990696838 0.191058236556442
1.99652919147221 0.215505887700011
1.99459957647011 0.140056664895674
1.99806054285466 0.120547231379614
1.98593862830166 0.0916486389328984
1.97931641143295 0.462734428071515
1.99855631180132 0.101120011114003
1.99421206141539 0.258443908859530
1.99781790486039 0.543516910843497
1.99720522726900 0.0737173931186571
1.98303440848516 0.152622777636722
1.99900862513681 0.674526132004626
1.99866761178096 0.358292746434059
1.99783983352391 0.491305146804456
1.99960825876476 0.122579254402338
1.96710953562570 0.0513811784835662
1.99842776886770 0.0112006869294710
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