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There is no closed-form solution, but solving for $r^\star$ such that $$f(r^\star) = \tilde{c}^{-1} $$ should be fast and safe with a standard single dimension solver, bisection or Newton-Raphson, as function $f$ is monotonically decreasing ($B_i$'s and $\tilde{c}_i$ are positive), $$f(x) = \sum_{i=1}^n \tilde{c}_i {\rm e}^{A_i-B_ix}, $$ its derivative is ...


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Given the non-linear nature of the constrained optimization problem ie. $exp(A(T0,Ti)-B(T0,Ti)*r)$, you will need to employ numerical solvers. The authors of the document used Simulated Annealing (shown in Appendix B) for fast convergence. They note that it could take up to 10 seconds to solve a 10-dimensional parameter space.


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The forwards and the spot rates will be decreasing, that is correct.


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