A new polynomial-time algorithm for linear programming
Combinatorica
A polynomial-time algorithm, based on Newton's method, for linear programming
Mathematical Programming: Series A and B
On the Power of Real Turing Machines over Binary Inputs
SIAM Journal on Computing
An exact duality theory for semidefinite programming and its complexity implications
Mathematical Programming: Series A and B
Complexity and real computation
Complexity and real computation
A mathematical view of interior-point methods in convex optimization
A mathematical view of interior-point methods in convex optimization
Introduction to the Theory of Computation
Introduction to the Theory of Computation
On the Complexity of Semidefinite Programs
Journal of Global Optimization
On the Power of Random Access Machines
Proceedings of the 6th Colloquium, on Automata, Languages and Programming
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
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We address the exact semidefinite programming feasibility problem (SDFP) consisting in checking that intersection of the cone of positive semidefinite matrices and some affine subspace of matrices with rational entries is not empty. SDFP is a convex programming problem and is often considered as tractable since some of its approximate versions can be efficiently solved, e.g. by the ellipsoid algorithm. We prove that SDFP can decide comparison of numbers represented by the arithmetic circuits, i.e. circuits that use standard arithmetical operations as gates. Our reduction may give evidence to the intrinsic difficulty of SDFP (contrary to the common expectations) and clarify the complexity status of the exact SDP-an old open problem in the field of mathematical programming.