The 2013 Newton Institute Programme on polynomial optimization
The rapidly growing field of polynomial optimisation (PO) is concerned with optimisation problems in which the objective and constraint functions are all polynomials. There are applications of PO in a surprisingly wide variety of contexts, including, for example, operational research, statistics, applied probability, quantitative finance, theoretical computer science and various branches of engineering and the physical sciences. Not only that, but current research on PO is remarkably interdisciplinary in nature, involving researchers from all of the abovementioned disciplines, together with several branches of mathematics including graph theory, numerical analysis, algebraic geometry, commutative algebra and moment theory.
This special issue of Mathematical Programming Series B was originally conceived during a 4week residential programme on PO which took place in July and August 2013 at the Isaac Newton Institute for the Mathematical Sciences, an internationally recognised research institute in Cambridge, United Kingdom. The programme included a summer school, a workshop, a series of presentations from key speakers, and a day devoted to interactions between academics and industry. The organizers of the event were Adam N. Letchford, Jean Bernard Lasserre, Markus Schweighofer and Jörg Fliege, with Monique Laurent and Kurt Anstreicher acting as international advisors. The event was attended by over 80 researchers from over 20 countries. Details of the programme can be found on the web at: http://www.newton.ac.uk/event/POP.

van Dam and Sotirov give some new bounding procedures for a rather general graph partitioning problem, and show that the resulting bounds dominate previous known ones, yet remain tractable in practice.

Peña et al. show that a wide family of PO problems can be reformulated as linear optimisation problems over completely positive cones. This extends a result of Burer, which applied only to certain mixed 0–1 quadratic programs.

de Klerk et al. analyse a known polynomialtime approximation scheme for minimising a polynomial of fixeddegree over the simplex. They establish intriguing connections with Bernstein polynomials and multinomial distributions.

Bomze and Overton study a trustregionlike problem in which one wishes to optimise a nonconvex quadratic function over the intersection of two ellipsoids. They give conditions for this problem to be polynomialtime solvable.

Magron et al. use a combination of maxplus algebra and sumsofsquares arguments to compute verifiable numerical proofs of validity for inequalities involving polynomials and/or transcendental functions.

Lasserre studies certain convex sets that are defined via polynomial inequalities and quantifiers. These have applications in robust optimisation. He shows how to construct tractable inner and outer approximations of these sets.

de Laat and Vallentin presents an innovative approach to packing problems in discrete geometry, based on a (highly nontrivial) combination of graph theory, topology and semidefinite programming.

Nie uses a combination of convex algebraic geometry and semidefinite programming to derive local optimality conditions for PO problems. In some cases, these conditions enable one to enumerate local minima efficiently.

Ottem et al. use convex algebraic geometry to classify all convex sets that can be obtained by intersecting the semidefinite cone of order four with a threedimensional hyperplane.

Gouveia et al. use matrix factorisations and second order cone constraints to construct convex sets which lie in a space of polynomial dimension and which, when projected into an appropriate subspace, approximate any given polytope.

Djeumou Fomeni et al. present a new approach to mixed 0–1 polynomial programs, based on a combination of the classical ReformulationLinearisation Technique and a new way of generating cutting planes.

Vera and Dobre consider PO problems whose difficulty arises due to their extreme symmetry. They show that approaches based on copositive representations can be speeded up significantly by handling the symmetry explicitly.
We wish to thank the former and current directors of the Newton Institute (Sir David Wallace and Professor John Toland) for allowing the PO programme to take place, along with all of the administrative staff at the Institute, who enabled the event to run smoothly. We also thank the former and current editors of MPB (Professor Danny Ralph and Professor JongShi Pang) for permitting and overseeing the special issue, along with all of the authors who submitted papers to the issue, and all of those who acted as anonymous referees.