# Applied Mathematics

## Infinite-Dimensional linear systems

Existence of solutions.

We consider cone-constrained linear systems in Banach spaces, whose positive cone is induced by a partial ordering. We have provided a very general Farkas-like Theorem of the alternative which avoids the restrictive closedness assumption on the image of the cone by the linear mapping.
This permits to derive necessary and sufficient conditions for existence of solutions to many problems such as Volterra or Fredholm equations (with non-compact operators), the probabilistic Poisson equation, etc ... In the case of "Banach lattices", the conditions have an even simpler form.
Computational Aspects

We consider a general linear programming problem in some Banach space and provide a (numerical) iterative scheme for computing the optimal value. We approximate the infinite-dimensional LP problem by a sequence of finite-dimensional LPs and prove (under weak assumptions) the asymptotic convergence of the corresponding sequence of optimal values to the optimal value of the original problem.

We also prove that every accumulation point (in some appropriate weak topology) of the sequence of optimal solutions is an optimal solution of the original problem.
This also permits to derive a procedure for solving some linear equations, e.g. for computing the invariant probability of a Markov chain, for solving some linear partial diffential equations, etc ... To our knowledge, this is the first general computational scheme with such weak assumptions.

## Integration on a polytope

We consider the (computational) problem of integrating a continuous function on a polytope in \$R^n\$. We have shown that integrating a homogeneous function reduces to integrating the same function on the boundary, i.e. the faces of the polytope. Therefore, integrating a polynomial reduces to evaluations of the functions and its derivatives at the vertices only.
We had already used a similar property to compute the volume of a polytope, with good performances. We have extended some of the previous results to arbitrary compact domains. We have also proposed another approach. We consider the result as a function of the right-hand-side and obtain a simple closed form expression of its Laplace transform. We then invert the latter by Cauchy residues techniques. For papers citing our work see the section ``Softwares".

## Legendre-Fenchel duality and homogeneous functions

We have shown that homogeneous functions play a special role with repect to the Legendre-Fenchel transform in "Convex Analysis". Indeed, the p-homgeneous functions are changed into q-homogeneous fucntions with 1/p+1/q=1. The same result is also true in other algebras for the corresponding transforms, e.g. the Bellman-Karush transform in the "min,+" algebra, the Laplace-transform in the "+,." algebra, the "min,max" transform in the "min,max" algebra.

Singular perturbations of linear operators

We are interested  in the singular perturbation of linear operators like the fundamental matrix of Markov chains, pseudo-inverse operators, Sylvester and Lyapunov equations, etc. The ultimate goal is to provide a simple procedure to get the Laurent series expansion in terms of the perturbation parameter. This Laurent series expansion provides insight on the bad behavior of the operator for small values of the parameter. See for instance the paper Laurent series based RBF-FD method to avoid ill-conditioning. P. Gonzalez-Rodriguez, V. Bayona and M. Moscoso. Engineering Analysis with Boundary Elements 52 (2015), pp. 24--31.

Global optimization: Theory of moments and positive polynomials.

We are interested in the application of the theory of moments to global optimisation involving polynomials. We have been a pioneer in the development of what we call the Moment-SOS approach (see also the Lasserre hierarchy) This approach is very natural and the resulting LMI relaxations perfectly match both sides of the same theory, namely the theory of moments and its dual theory of positive polynomials. As a by-product one also obtains Karush-Kuhn-Tucker global optimality conditions, in which the multipliers are nonnegative polynomials instead of scalars as in the usual (local) optimality conditions. To appreciate the impact of this methodology in Optimization, see for instance:

- Optimization over polynomials: Selected Topics, M. Laurent, Proceedings of ICM conference, Seoul, 2014.
and especially for its impact in Combinatorial Optimization and Theoretical Computer Science, see for instance:
- Sum-of-Squares proofs and the quest toward optimal algorithms, B. Barak and D. Steurer, Proceedings of ICM conference, Seoul, 2014.
- Candidate Lasserre Integrality Gap For Unique Games, S. Khot and D. Mohskovitz, Electronic Colloquium on Computational Complexity, Report No. 142 (2014).

In particular and recently, this methodology has been used:
- for Motion Planning in Robotics. See the algorithm NUROA ( Numerical Roadmap Algorithm) and the associated paper:
NUROA: A Numerical Roadmap Algorithm, Reza Iraji, Hamidreza Chitsaz. Proceedings IEEE CDC Conference, Los Angeles, December 20214. arXiv:1403.5384
- For solving the Optimal Power Flow Problem, an important problem for the optimal management of energy networks. See e.g. the recent papers:
1. Advanced optimization methods for power systems, P. Panciatici, M.C. Campi, S. Garratti, S.H. Low, D.K. Molzhan, A.X. Sun and L. Wehenkel. Proceedings 18th Power Systems Computation Conference, Wroclaw, Poland, August 2014.
2. Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem, D.K. Molzahn and I.A. Hiskens, IEEE Transactions on Power Systems, 2015. arXiv:1404.5071
3. Global Optimization for power dispatch problems based on theory of moments, J. Tian, H. Wei and J. Tan, Electrical Power and Energy Systems 71, 2015, pp. 184--194.

This approach also permits to obtain (numerically) bounds on measures that satisfy pre-specified moment conditions. As a consequence we can apply these techniques to provide upper and lower bounds in several problems of performance evaluation. For instance, for option pricing models in financial mathematics based on diffusions (such as Black and Scholes, Ornstein-Ulhenbeck, Cox-Ingersoll interest models) we may obtain good  upper and lower bound on e.g. Asian and Barrier options. The method is generic to diffusion type models.

More generally, we are also interested in all aspects of positivity of polynomials (and even semi-algebraic functions) on basic semi-algebraic sets, including certificates of positivity with tractable characterization (to be amenable to practical computation).

Discrete optimization and counting

We are interested  in the problem of counting lattice points in a convex polytope with applications to discrete optimization. The idea is to use generating functions in the spirit of Brion and Vergne, Kantor, Sinai and Robins. We thus work in the space of "dual" variables associated with the nontrivial constraints that define the polytope; we then use the Z-transform (or generating function) of the function that counts the lattice points, considered as a function of the right-hand-side of the constraints. We then develop techniques to invert the Z-transform. We also use Brion and Vergne's periodic formula to derive a duality for integer programs, that parallels that of linear programming. In the same vein, we have also obtained a discrete Farkas lemma (for existence of a nonnegative integral solution to the linear system Ax=b) as well as an algebraic characterization of the integer hull of the convex rational polytope {Ax=b; x >=0}.

Inverse problems

We are also interested in some Inverse Problems of Computational Geometry, Optimization, and Optimal Control. Namely:
- Reconstructing an unknown n-dimensional geometric object from the only knowledge of moments of a measure supported on this object.
- Inverse optimization: Given a feasible solution "y" and a criterion "f" to minimize, find a criterion "g" as close as possible to "f" and such that "y" is a global optimizer for this new criterion.
- Inverse Optimal Control: Given a systems dynamics and a database of trajectories, find a Lagrangian for which those trajectories are optimal.