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⛄ 内容介绍
Portfolio selection is one of the major capital allocation and budgeting issues in financial management, and a variety of models have been presented for optimal selection. Semi-variance is usually considered as a risk factor in drawing up an efficient frontier and the optimal portfolio. Since semi-variance offers a better estimation of the actual risk portfolio, it was used as a measure to approximate the risk of investment in this work. The optimal portfolio selection is one of the non-deterministic polynomial(NP)-hard problems that have not been presented in an exact algorithm, which can solve this problem in a polynomial time. Meta-heuristic algorithms are usually used to solve such problems. A novel hybrid harmony search and artificial bee colony algorithm and its application were introduced in order to draw efficient frontier portfolios. Computational results show that this algorithm is more successful than the harmony search method and genetic algorithm. In addition, it is more accurate in finding optimal solutions at all levels of risk and return.
⛄ 部分代码
is a simple demo version implementing the basic Harmony Search %
% algorithm for clustering problem without fine-tuning the parameters. %
% Though this demo works very well, it is expected that this illustration %
% is much less efficient than the work reported in paper: %
% (Citation details): %
% J. Senthilnath, Sushant Kulkarni, Raghuram, D.R., Sudhindra, M., %
% Omkar, S.N., Das, V. and Mani, V (2016) "A novel harmony search %
% based approach for clustering problems", Int. J. Swarm Intelligence, %
% Vol. 2, No. 1, pp.66 ?86, 2016. %
%%-------------------------------------------------------------------------%function [BestGen] = Harmony_Search(traindat,limits,attr)
% Harmony search on illustrative dataset
% traindat----> matrix containing dataset without class labels (nxp)
% UL ----> matrix with upper and lower limits for "p" attributes (px2)
% HMCR---> Harmony Memory Consideration Rate [HMCR_max HMCR_min]
% HMS----> Harmony Memory Size or population size
% NVAR---> Number of attributes (p)
% PAR---> Pitch Adjustment Rate [PAR_max PAR_min]
global ULB HMS HMCR PAR_max PAR_min BW_min BW_max tem ;
global BestFit WorstFit BestGen BW HM fitness;
global counter BestIndex WorstIndex attributes;
ULB = limits;
BestGen =[]; % Store the best fitness
HMS = 5; % Size of population
NATTR = attr; % No. of attributes in datase
MaxIter = 5000; % Maximum no. of iterations
HMCR_max = 0.95; HMCR_min = 0.06;
PAR_max = 0.95; PAR_min = 0.35;
BW_max = 0.01; BW_min=0.1;
% Initialize Matrices
BW = zeros(1,NATTR); % Bandwidth values [1xp]
HM = zeros(HMS,NATTR) ; % Size of Harmony Memory (HMSxp)
fitness = zeros(1,HMS); % Fitness of each population
attributes = zeros(1,NATTR) ;
tem = zeros(2,NATTR) ;
HarmonySearch;
% plot the final optimal cluster center
plot(BestGen(1),BestGen(2),'k*', 'LineWidth',3)
legend('Agents','Agents Movement','Location','NorthEastOutside')
xlabel('X'); ylabel('Y'); xlim([0 30]); ylim([0 35]);
text(35,30,'* Cluster Centers', 'FontName','Times','Fontsize',12)
% plot(BestGen(1),BestGen(2),'g*') % plot the final optimal cluster center
% /***********************************************************************/
function objective = f(x)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OBJECTIVE FUNCTION - Evaluation of fitness value of each solution vector
% x receives the matrix containing the pseudo optimal cluster centres to be evaluated
% Optimisation of Clustering Criteria by Reformulation
% Refer Equation (4) in paper: "Optimisation of Clustering Criteria by Reformulation
% Note: your own objective function can be written here
% when using your own function, remember to change limits/bounds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
objective = sum(pdist2(x,traindat));
end
% /***********************************************************************/
% Initialisation of Harmony Memory with random solutions between limits
function initialize
for i = 1:HMS
for j = 1:NATTR
% Initialization of HM between upper and lower limit
HM(i,j) = ((ULB(j,2)-ULB(j,1))*rand(1))+ULB(j,1);
end
fitness(i)= f(HM(i,:)); % evaluate fitness of each HM
end
% plot initial solutions for visualization
plot(HM(:,1),HM(:,2),'gs', 'LineWidth',1.5);
hold on;
end
% /***********************************************************************/
% Improvisation of Harmony Memory
function HarmonySearch
initialize; % First intialize HM with random values
counter = 0; % Loop counter
while counter < MaxIter
% dynamically vary the Harmony Search parameters - PAR, HMCR, bw
PAR=(PAR_max-PAR_min)/(MaxIter*counter)+PAR_min;
coef=log(BW_min/BW_max)/MaxIter;
for pp =1:NATTR
BW(pp)=BW_max*exp(coef*counter);
end
for hh = 1:NATTR
HMCR(hh) = HMCR_min+(((HMCR_max-HMCR_min)/MaxIter)*counter);
end
% improvise the solutions using parameters
for i = 1:NATTR
hmcr_rnd = rand(1);
if hmcr_rnd <= HMCR(i)
index = randi([1,HMS],1); % randomly pick one population from HM
attributes(i) = HM(index,i);
par_rnd = rand(1);
% if random number is less than PAR, tune solution
if par_rnd <= PAR
walk_rnd = rand(1);
temp = attributes(i);
if walk_rnd<0.5 % tune i.e. add/subtract small value with 50% probabilty
temp = temp+rand(1)*BW(i);
% check if improvised solution is within upper limit
if temp < ULB(i,1)
attributes(i) = temp;
end
else
temp = temp-rand(1)*BW(i);
% check if improvised solution is within lower limit
if temp>ULB(i,2)
attributes(i) = temp;
end
end
end % End of if for r2
else
% Randomisation with probability of 1-HMCR
attributes(i) = ((ULB(i,1)-ULB(i,2))*rand(1))+ULB(i,2);
end
end
value = f(attributes); % calculate fitness of improvised solution
UpdateHM(value); % Update the fitness value of the HM
counter = counter+1; % Increment loop counter
end
end
% /***********************************************************************/
% Update the Harmony Memory
function UpdateHM( NewFit )
% For the first iteration, find best & worst fitness solutions with index
if(counter==0)
[BestFit, BestIndex] = min(fitness);
[WorstFit, WorstIndex] = max(fitness);
end
% for second iteration iteration onwards
tem(1,:) = HM(WorstIndex,:);
% replace if less than worst fitness
if (NewFit < WorstFit)
% if lower than previous best fitness, update new fitness
% as the best fitness while discarding the worst fitness
if( NewFit < BestFit )
HM(WorstIndex,:)=attributes;
BestGen=attributes;
tem(2,:) = attributes;
fitness(WorstIndex)=NewFit;
BestIndex=WorstIndex;
% if only lower than worst fitness but not best fitness, then just
% replace worst fitness
else
HM(WorstIndex,:)=attributes;
tem(2,:) = attributes;
fitness(WorstIndex)=NewFit;
end
% plot the movement of the solutions
pause(0.0005) % Pause interval for tracing movements can be changed here
hold on;
plot(tem(:,1),tem(:,2),'k:');
% find the new worst solution and its index
[WorstFit, WorstIndex] = max(fitness);
end % NewFit if
end % function update
toc;
end % end of function
⛄ 运行结果
⛄ 参考文献
[1]戈国华, 肖海波, 张敏. 基于FCM的数据聚类分析及Matlab实现[J]. 福建电脑, 2007(4):2.
[2] Seyedhosseini, S. M. , et al. "A novel hybrid algorithm based on a harmony search and artificial bee colony for solving a portfolio optimization problem using a mean-semi variance approach." Journal of Central South University (2016).
