The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.

We study numerical stability for interior-point methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.

Let f be a polynomial in Q[X1,..., Xn] of degree D. We provide an efficient algorithm in practice to compute the global supremum supx∈Rn f(x) of f (or its infimum inf x∈Rn f(x)). The complexity of our method is bounded by D O(n). In a probabilistic model, a more precise result yields a complexity bounded by O(n 7 D 4n) arithmetic operations in Q. Our implementation is more efficient by several orders of magnitude than previous ones based on quantifier elimination. Sometimes, it can tackle problems that numerical techniques do not reach. Our algorithm is based on the computation of generalized critical values of the mapping x → f(x), i.e. the set of points {c ∈ C | ∃(xℓ)ℓ∈N ⊂ C n f(xℓ) → c, ||xℓ||||dxℓf| | → 0 when ℓ → ∞}. We prove that the global optimum of f lies in its set of generalized critical values and provide an efficient way of deciding which value is the global optimum.

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