# Defining Non Linear Vector Constraints in JuMP

I’m trying to implement pseudospectral methods using JuMP & Ipopt. The state is n-dimensional and control m-dimensional and the values are evaluated at N+1 collocation points; in this particular problem, n=3, m=3, and N = 120.

For pseudospectral methods, one constraint is that the numerical approximation of the derivative 2/TDX = f_all(X), where X is the stacked state [x1;x2…;x_{N+1}] and f_all = [f(x1); f(x2);…;f(x_{N+1})].

@NLconstraint(mod, f_all(dyn, state, N, n, m) - 2/Tp*D*state[1:n*(N+1)] == zeros(n*(N+1)))

dyn is the non-linear dynamics function and f_all returns a stacked vector of dyn() evaluated for each x.

dyn(p,w) = 0.25*((1-norm(p)^2)*w - 2*skew(w)*p + 2*dot(w,p)*p);
JuMP.register(mod, :dyn, 2, dyn, autodiff=true);

I keep running into numerous errors regardless of how I try to define the nonlinear functions:

ERROR: Cannot multiply a quadratic expression by a variable

I also tried breaking apart the n-D derivative into n different @NLexpression:

@NLexpression(mod, dyn1_expr[1:n], [0.25*(p[1]^2-p[2]^2-p[3]^2+1)*w[1] +  0.5*(p[1]*p[2]-p[3])*w[2] + 0.5*(p[2]-p[1]*p[3])*w[3]; 0.5*(p[1]*p[2]+p[3])*w[1] + 0.25*(p[2]^2 - p[1]^2-p[3]^2)*w[2] + 0.5*(p[2]*p[3]-p[1])*w[3]; 0.5*(p[1]*p[3]-p[2])*w[1] + 0.5*(p[1]+p[2]*p[3])*w[2] + 0.25*(p[3]^2-p[2]^2-p[1]^2+1)*w[3]])

ERROR: Unexpected vector JuMP.GenericQuadExpr{Float64,JuMP.Variable}[1.736231682755747e-5 state[1]² + 0.00033454250210412234 state[1]*state[2] - 1.736231682755747e-5 state[2]² - 0.0006879201991741926 state[1]*state[3] - 1.736231682755747e-5 state[3]² - 0.00033454250210412234 state[3] + 0.0006879201991741926 state[2] + 1.736231682755747e-5,-0.00016727125105206117 state[1]² + 3.472463365511494e-5 state[1]*state[2] + 0.00016727125105206117 state[2]² + 0.0006879201991741926 state[2]*state[3] - 0.00016727125105206117 state[3]² + 3.472463365511494e-5 state[3] - 0.0006879201991741926 state[1],-0.0003439600995870963 state[1]² - 0.0003439600995870963 state[2]² + 3.472463365511494e-5 state[1]*state[3] + 0.00033454250210412234 state[2]*state[3] + 0.0003439600995870963 state[3]² - 3.472463365511494e-5 state[2] + 0.00033454250210412234 state[1] + 0.0003439600995870963] in nonlinear expression. Nonlinear expressions may contain only scalar expressions.

Any idea how I could go about defining an n*(N+1) dimensional constraint by evaluating the dynamics f(x) N+1 times?

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JuMP’s syntax and automatic differentiation utilities were not designed to handle this case. There’s a workaround for multidimensional input (Julia+JuMP: variable number of arguments to function - Stack Overflow) but not for multidimensional output. I’d recommend looking into tools like CasADi, or you can use ReverseDiff to compute derivatives and manually communicate them to Ipopt via MathProgBase.

If I were to break apart the dynamics function f into n separate functions, would that circumvent the issue? Since I know the matrix D from D*X, I could grab the i’th row and call a function f_i that also returns the i’th output of f(x). It seems inefficient, but that would allow me to pass in an n dimensional input and return one output at a time.

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Yes, you can do that. I won’t claim that it’s pretty though.

Is it still the case that matrix-vector products cannot be used in nonlinear constraints?

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@acauligi, If you don’t mind . If you got any working solution to this problem, kindly provide it here. Thank you.