Applications of Convex.jl in Optimization Involving Complex Numbers slides
About Me¶
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BS and MS in Mathematics and Computing Sciences (July'12-June'17) at Indian Institute of Technology Kharagpur
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Google Summer of Code 2016 and 2017 student under the Julia Language
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GitHub: Ayush-iitkgp
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Website: https://ayush-iitkgp.github.io
Outline¶
- Power Flow Optimization
- Fidelity in Quantum Information
Optimal Power Flow - Introduction¶
- Power flow study is an analysis of a connected electrical power system’s capability to adequately supply the connected load.
- Unknowns: - Voltages angle and magnitude information for each bus
- Knowns: - Load( such as appliances and lights.) and generator real power and voltage condition.
Optimal Power Flow - Mathematical Formulation¶
Constraints¶
- $p_{ij}$: Active power entering the line from node i
- $q_{ij}$: Reactive power entering the line from node i
- Let $x_{i}$ denote the complex voltage for node i of the network.
We have the following power balance equations which are non-linear in unknown $x_{i}$ and $x_{j}$.
Objective¶
Depends upon the business needs such as:
- Minimize power losses in an electrical network
- Minimize cost of generation
Optimal Power Flow - SDP Relaxation¶
- The original optimal power flow problem is non-convex in nature.
- Thanks to the lifting technique which converts the above optimization problem to a SemiDefinite Programming Problem.
- The relaxed SDP problem finds the near global solution of the original non-convex problem.
Example¶
- The data is taken from the IEEE 14 Bus test case which represents a portion of the American Electric Power System (in the Midwestern US) as of February, 1962.
using Convex # Read the input data
using FactCheck
using MAT #Pkg.add("MAT")
TOL = 1e-2;
input = matopen("Data.mat")
varnames = names(input)
Data = read(input, "inj","Y");
n=size(Data[2],1); # Create some intermediate variables
Y=Data[2];
inj=Data[1];
W = ComplexVariable(n,n); # W is the matrix of pairwise products of the voltages
objective = real(sum(diag(W))); # The objective is to minimize cost of generation
c1 = Constraint[]; # The constraints are power balance equations
for i=2:n
push!(c1,sum(W[i,:].*(Y[i,:]'))==inj[i]);
end
c2 = W in :SDP;
c3 = real(W[1,1])==1.06^2;
push!(c1, c2);
push!(c1, c3);
p = maximize(objective,c1); # Create the problem
solve!(p); # Solve the problem
p.optval #15.125857662600703
evaluate(objective) #15.1258578588357
output = matopen("Res.mat"); # Verify the results
names(output);
outputData = read(output, "Wres");
Wres = outputData;
real_diff = real(W.value) - real(Wres);
imag_diff = imag(W.value) - imag(Wres);
@fact real_diff => roughly(zeros(n,n), TOL);
@fact imag_diff => roughly(zeros(n,n), TOL)
Fidelity in Quantum Information Theory - Introduction¶
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This example is inspired from a lecture of John Watrous in the course on Theory of Quantum Information.
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Fidelity is a measure of the closeness of two quantum states.
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The ability to distinguish between the quantum states is equivalent to the ability to distinguish between the classical probability distributions.
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If fidelity between two states is 1, they are the same quantum state.
- Application
- The Fidelity between two Hermitian semidefinite matrices P and Q is defined as:
where the trace norm $||.||_{tr}$ is the sum of the singular values, and the maximization goes over the set of all unitary matrices U.
Fidelity in Quantum Information Theory - Mathematical Formulation¶
Fidelity can be expressed as the optimal value of the following complex-valued SDP:
$$ \textbf{maximize} \frac{1}{2} trace(Z+Z^*)$$$$\text{subject to } \left[\begin{array}{cc}P&Z\\{Z}^{*}&Q\end{array}\right] \succeq 0$$$$\text{where } Z \in \mathbf {C}^{n \times n}$$Example¶
n = 20 # Create the data
P = randn(n,n) + im*randn(n,n);
P = P*P';
Q = randn(n,n) + im*randn(n,n);
Q = Q*Q';
Z = ComplexVariable(n,n); # Declare convex variable
objective = 0.5*real(trace(Z+Z')); # Specify the objective
constraint = [P Z;Z' Q] ⪰ 0;
problem = maximize(objective,constraint);
solve!(problem) # Solve the problem
# Verify that computer fidelity is equal to actual fidelity
computed_fidelity = evaluate(objective)
P1,P2 = eig(P);
sqP = P2 * diagm([p1^0.5 for p1 in P1]) * P2'
Q1,Q2 = eig(Q)
sqQ = Q2 * diagm([q1^0.5 for q1 in Q1]) * Q2'
actual_fidelity = sum(svd(sqP * sqQ)[2])
diff = computed_fidelity - actual_fidelity
@fact diff => roughly(0, TOL)