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**11 - 12**of**12**### ON VERIFIED NUMERICAL COMPUTATIONS IN CONVEX PROGRAMMING

"... Abstract. This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analy ..."

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Abstract. This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations to special types of convex problems and convex relaxations is included. The latter are important for handling and for reliability issues in global robust and combinatorial optimization. Some remarks on numerical experiences, including also large-scale and ill-posed problems, and software for verified computations concludes this survey. Key words. Linear programming, semidefinite programming, conic programming, convex programming, combinatorial optimization, rounding errors, ill-posed problems, interval arithmetic, branch-bound-and-cut AMS subject classifications. primary 90C25, secondary 65G30 1. Introduction. Mathematical

### Noname manuscript No. (will be inserted by the editor) On Rigorous Upper Bounds to a Global Optimum

"... Abstract In branch and bound algorithms in constrained global optimization, a sharp upper bound on the global optimum is important for the overall efficiency of the branch and bound process. Software to find local optimizers, using floating point arithmetic, often computes an approximately feasible ..."

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Abstract In branch and bound algorithms in constrained global optimization, a sharp upper bound on the global optimum is important for the overall efficiency of the branch and bound process. Software to find local optimizers, using floating point arithmetic, often computes an approximately feasible point close to an actual global optimizer. Not mathematically rigorous algorithms can simply evaluate the objective at such points to obtain approximate upper bounds. However, such points may actually be slightly in-feasible, and the corresponding objective values may be slightly smaller than the global optimum. A consequence is that actual optimizers are occasionally missed, while the algorithm returns an approximate optimum and corresponding approximate optimizer that is occasionally far away from an actual global optimizer. In mathematically rig-orous algorithms, objective values are accepted as upper bounds only if the point of evaluation is proven to be feasible. Such computational proofs of feasibility have been weak points in mathematically rigorous algorithms. This paper first reviews previously proposed automatic proofs of feasibility, then proposes an alternative technique. The alternative technique is tried on a test set that caused trouble for previous techniques, and is also employed in a mathematically rigorous branch and bound algorithm on that test set.