In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper-and the lower-level problems are polynomials. We present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations, whereas our approach for general problems involving a nonconvex polynomial lower-level problem solves a sequence of approximation problems via another sequence of SDP relaxations.

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).

The trust-region problem,which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semialgebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdos raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). Given a graph $H$, what is the maximum number of edges of a graph with $n$ vertices not containing $H$ as a subgraph? This number is denoted $ex(n,H)$, and is known as the Turan number. Erdos conjectured that there exists a positive constant $c$ such that $ex(n,C_{2k})\geq cn^{1+1/k}$(see P. Erdos, Some unsolved problems in graph theory and combinatorial analysis, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) , pp. 97--109, Academic Press, London, 1971). Haj\'{o}s conjectured that every simple even graph on $n$ vertices can be decomposed into at most $n/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp. 231 - 236 ). We present the problems, conjectures related to these problems and we summarize the know results.