Computing approximations and generalized solutions using moments and positive polynomials

The generalized moment problem (GMP) is a linear optimization problem over spaces of measures. It allows to model many challenging mathematical problems. While in general it is impossible to solve the GMP, in the case where all data are polynomial and semialgebraic sets, one can define a hierarchy of semidefinite relaxations - the moment-sums-of-squares (moment-SOS) hierachy - which in principle allows to approximate the optimal value of the GMP to arbitrary precision. The work presented in this thesis addresses two facets concerning the GMP and the moment-SOS hierarchy: One facet is concerned with the scalability of relaxations for the GMP. The degree of the SOS weights in the moment-SOS hierarchy grows when augmenting the relaxation order. When the number of variables is not small, this leads quickly to semidefinite programs (SDPs) that are out of range for state of the art SDP solvers, unless one can use symmetries or some structured sparsity which is typically present in large scale applications. We provide a new certificate of positivity which (i) is able to exploit the structured sparsity and (ii) only involves SOS polynomials of fixed degree. From this, one can define a new hierarchy of SDP relaxations for the GMP which can take into account sparsity and at the same time prevents from explosion of the size of SDP variables related to the increasing degree of the SOS weights in the standard hierarchy. The second facet focusses on (i) modelling challenging problems as a particular instance of the GMP and (ii) solving these problems by applying the moment-SOS hierarchy. In particular we propose deterministic approximations of chance constraints a difficult problem as the associated set of feasible solutions is typically non-convex and sometimes not even connected. In our approach we replace this set by a (smaller) sub-level-set of a polynomial whose vector of coefficients is a by-product of the moment-SOS hierarchy when modeling the problem as an instance of the GMP. The quality of this inner approximation improves when increasing the degree of the SDP relaxation and asymptotic convergence is guaranteed. The procedure is illustrated by approximating the feasible set of an instance of the chance-constrained AC Optimal Power Flow problem (a nonlinear problem in the management of energy networks) which nowadays becomes more and more important as we rely increasingly on uncertain energy sources such as wind and solar power. Furthermore, we propose an extension of this framework to the case where the underlying distribution itself is uncertain and provide guarantees of convergence. Another application of the moment-SOS methodology discussed in this thesis consider measure valued solutions to optimal control problems. We show how this procedure can capture the limit behavior of an optimizing sequence of control and its corresponding sequence of trajectories. In particular we address the case of concentrations of control and discontinuities of the trajectory may occur simultaneously. In a final contribution, we compute measure valued solutions to scalar hyperbolic conservation laws, such as Burgers equation. It is known that this class of nonlinear partial differential equations has potentially discontinuous solutions which are difficult to approximate numerically with accuracy. Under some conditions the measure valued solution can be identified with the classical (weak) solution to the conservation law. In this case our moment-SOS approach provides an alternative numerical scheme to compute solutions which in contrast to existing methods, does not rely on discretization of the domain.