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1538 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 the visibility at X i . The basic framework of BIQACA described as follows: 1) Selecting the target position of ant movement By applying the principle of randomness, a number of qubits in the current position were randomly selected to constitute a position update vector S. The transition rule and transition probability of ant k from position X to position where q∈[0, 1] is even-distributed random number, q 0 ∈[0, 1] is probability parameter, P is the set of occupied points for ant in unit space, X s tion as per formula (8) ; α is the update parameter of pheromone, β is the update parameter of visibility. 2) Realizing the movement of ant towards target position via quantum rotation gate After the ant has selected the target position, its movement process can be realized by changing the phase of qubit it brought for quantum rotation gate. In unit space, suppose the current position for ant at time t is P i , selected target position is P k , update vector of P i is S, then the update of phase angle increment at P i is where )} t+ 1 t+ 1 t t t t+ 1 t t+ 1 ⎡cosϕij sinθ ⎤ ⎡ ij cosϕij sinθ ⎤ ⎡ ij cos( ϕij +Δ ϕij )sin( θij +Δθ ) ⎤ ij ⎢ t+ 1 t+ 1⎥ ⎢ t t ⎥ ⎢ t t+ 1 t t+ 1 ⎥ ⎢sinϕij sinθij ⎥ = U ⎢sinϕij sinθij ⎥ = ⎢sin( ϕij +Δ ϕij )sin( θij +Δθij ) ⎥ ⎢ t+ 1 t t t 1 cosθ ⎥ ⎢ ij cosθ ⎥ ⎢ + ij cos( θij θ ) ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ +Δ ij ⎦ (15) Apparently, U-gate can rotate the phase of qubit by t 1 ϕ + t 1 Δ ij and Δ θ + ij . 3) Adjustment strategy of search space i In BIQACA, the search space for each qubit is designed as [ lowBd ij , upBdij ] , the search space at initializa- X s are: α β ⎧arg max{ τ ( X s) iη ( Xs)} q ≤ q tion is [ 0.25π ,0.75π ] , during optimizing process of ants, 0 ⎪ Xs∈P X s = ⎨ these search spaces are related with the contraction level ⎪ ⎩ Xs q > q0 (8) of each qubit, and decrease exponentially, which can significantly improve the solution accuracy of the algorithm. α β τ ( Xs) iη ( Xs) t+ 1 t pX ( s ) = α β τ ( X u) η ( X u) X ∑ i , (9) ⎡lowBd ⎤ ⎡ 1 ⎤⎡ ij lowBd ⎤ ij ⎢ t+ 1 ⎥ = ⎢ t ⎥ ij ⎢ t ⎥ (16) nL ⎢ s, Xu∈P ⎣ upBdij ⎥⎦ ⎢⎣nf ⎥⎦⎢⎣upBdij ⎥⎦ where j ∈ { S(1), S(2), , S( sm)} , S is the update vector of ants, nf = 2 is the constriction factor, nL t represents ij is the selected target loca- the contraction level of t-th iteration. 4) Processing of ant position variation Suppose the current position is P i , update vector of P i is S, the search space of P i is [ lowBd ij , upBdij ] . Then the update of phase angle increment at P i is: ⎧ Δϕij = (2rand −1)( upBdij − lowBdij ) ⎨ ⎩ Δϕij = sign( Δϕij )( abs( Δϕij ) + lowBdij ) (17) ⎧ Δθij = (2rand −1)( upBdij − lowBdij ) ⎨ ⎩ Δθij = sign( Δθij )( abs( Δθij ) + lowBdij ) (18) ( φkj − φij ) × rand t ⎧⎪ j φkj ≠φij Δ φij = ⎨ (10) 5) Random behavior of ants ⎪⎩ Δ φij φkj = φij If P i is not improved after continuous limited-time cycles, the position should be abandoned, the ants will generate a new P ' t t ⎧Δ ϕij + 2π Δ ϕij < −π t+ 1 ⎪ i through random behavior to substtute P i . t t Δ ϕij = ⎨ Δ ϕij −π ≤ Δϕij ≤ π (11) ' ⎪ Δ t t ⎩ ϕij − 2π Δ ϕij > π ϕij = mean( ϕi ) + Δϕ ij (19) ' ( θkj − θij ) × rand t ⎧⎪ j θkj ≠θ θij = mean( θi ) + Δθ ij ij (20) Δ θij = ⎨ (12) ⎪⎩ Δ θij θkj = θij where i ∈ { 1,2, , n} , j ∈ { S(1), S(2), , S( sm)} , S is the update vector of current position, Δ ϕij and Δ θij are updated using (17), (18), mean( θ t t ⎧Δ θij + π Δ θij < −π /2 t+ 1 ⎪ t t i ) is the mean value of Δ θij = ⎨Δ θij −π /2 ≤ Δθij ≤ π /2 (13) vector of phase angle θ ⎪ Δ t t i at P i . ⎩ θij − π Δ θij > π /2 6) Update rules for pheromone intensity and visibility When the ant completes a traverse, the current position j ∈ { S(1), S(2), , S( sm , rand j is random num- is mapped into the solution space of optimal problem Δ ϕij , Δ θij can be obtained using from unit space, fitness function is calculated, and the intensity and visibility of pheromone at current position should be updated. ⎧τ( X i) = (1 − ρ) τ( Xi) + ρτ( Xi) t+ 1 t+ 1 t+ 1 t+ 1 t+ 1 t t+ 1 ⎨ ⎡cos Δϕij cos Δθij −sin Δϕij cos Δθij sin Δ θij cos( ϕij +Δϕij ) ⎤ ⎢ t+ 1 t+ 1 t+ 1 t+ 1 t t+ 1 ⎥ ⎩ τ ( X i) = Qfit( X i) (21) U = ⎢sin Δϕij cos Δθij cos Δϕij cos Δθij sin Δ θsin( ϕij +Δϕij ) ⎥ ⎢ t+ 1 t t+ 1 t+ 1 −sin Δθij −tan( ϕij / 2)sin Δθij cos Δθ ⎥ ⎣ ij ⎦ (14) ber between [0, 1]; (17), (18). Update of probability amplitude of qubit based on quantum rotation gate © 2013 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1539 η ( X ) = fit( X ) (22) i where (1 −ρ) ∈ [0,1] is the evaporation coefficient of pheromone, Q is the enhancement coefficient of pheromone. E. Description of BIQACA Taking the function extremum problem as an example, BIQACA implementation steps are described as follows: Step 1: Setup relevant parameters such as the number of ants, maximum number of iterations, number of limits limit, contraction level nL ij i , constriction factor nf, Maximum contraction level MaxL, reset contraction level resetL, search space [ lowBd ij , upBdij ] , etc. Step 2: Randomly generate initial position of ants according to (1), transform the solution space according to (2), and calculate the fitness of each ant according to (5) or (6). Update the pheromone intensity and visibility according to (21) and (22).Record the current optimum solution, i.e. global optimum solution GBest. Initialize the conceptual vector trial ( i) = 0 , record the number of nonupdates at the position of ant. Step 3: Update the search space [ lowBd ij , upBdij ] according to (16). Step 4: Select a moving target for each ant in the ant colony according to (8) and (9), then realize the movement of ants using quantum rotation gate in light of (10), (12) and (14). Step 5: For each ant, according to mutation probability, realize the variation of ant’s position using quantum rotation gate in light of (17) and (18). Step 6: Transform the solution space according to (2), calculate the fitness of each ant according to (5) or (6). Update the current position if the new position is better than the current one; otherwise trial ( i) = trial( i) + 1 , update contraction level nL ( i, j) = nL( i, j) + 1 , if nL ( i, j) >MaxL, nL ( i, j) =resetL. Step 7: Determine whether trial (i) is greater than the limit, if trial (i) >limit, abandon the current position of the i-th ant, and generate a new position according to (19), (20) and (15), perform space transformation in light of (2), calculate the fitness of each ant according to (5) or (6), trial ( i) = 0 . Step 8: Update the pheromone intensity and visibility according to (21) and (22). Record the current local optimum position, Best, and local worst position, Worst. Step 9: Determine whether the local optimum position, Best is greater than the global optimum position, GBest, if Best>GBest, update the global optimal position, GBest with local optimum position, Best, otherwise, update the local worst position, Worst with global optimum position, GBest. Step 10: Update the number of iterations t=t+1. If the current number of iterations t>maxgen or accuracy of convergence is met, stop the search, output the global optimum position, or else, turn to Step 3. III. SIMULATION EXPERIMENT To verify the effectiveness and feasibility of BIQACA algorithm, function extremum problem, traveling salesman problem and multicast routing problem were selected for testing. The simulation programs were programed and implemented in MATLAB 2009a, test results were obtained in a PC with an Intel Core (TM) i5 CPU running at 3.2GHz, and a 2.8GB RAM. A. Function Extremum Problem Three internationally commonly used functions f1~f3 were selected to test BIQACA performance when the number of independent variables was 2 and 30 respectively 2 2 f ( x, y) = 0.3cos3π x − 0.3cos 4πy − x − y − 0.3, −1 ≤ x, y 1 (23) 1 ≤ n f ( x ) = ∑ ( −x sin( x )), − 500 ≤ x 500 (24) 2 i i i i ≤ i= 1 n ∑ 2 f3( xi ) = ( xi −10cos(2π xi ) + 10), −5.12 ≤ xi ≤ 5. 12 (25) i= 1 1) When the number of independent variables is 2 Test function is f1, the optimization objective is to obtain the maximum value. Algorithm parameters: maximum number of iterations maxgen=500, number of ants n=20, probability parameter q 0 =0.5, evaporation coefficient 1− ρ =0.05, pheromone update parameter α =1, visibility update parameter β =5, pheromone enhancement coefficient Q =10, mutation probability P m =0.05; the program was terminated when the BIQACA algorithm found the optimal solution or had run for gen=500 iterations. Simulations were conducted using the CQACO algorithm and ACO algorithm in Ref. [12] respectively, each algorithm was run for 50 times independently under the same conditions, and their optimal value (Best), optimal mean value (M-best), number of success and average number of iterations were recorded, optimization results were compared in Table 1. TABLE I. COMPARISON OF EXPERIMENTAL RESULTS OF 3 ALGORITHMS WITH TEST FUNCTION F 1 Func/opt f 1/ 0.24 Status Algorithm ACO CQACO BIQACA Best 0.2400 0.2400 0.2400 M-best 0.1720 0.2388 0.2400 Con-times 42 48 50 Ave-Steps 198.16 84.04 17.06 It can be seen from Table 1 that, BIQACA algorithm’s optimization efficiency is the highest, its optimization results is also the greatest, with the success rate of 100%; followed is CQACO, with a success rate of 96%; the last is ACO, with a success rate of 84%. 2) When the number of independent variables is 30 Test functions were f2, f3, optimization goal is to obtain the minimum value. © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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