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Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... 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 primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 7 (2 self)
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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 primaldual 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.
Numerical Stability in Linear Programming and Semidefinite Programming
, 2006
"... We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms. ..."
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Cited by 1 (1 self)
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We study numerical stability for interiorpoint methods applied to Linear Programming, LP, and Semidefinite Programming, SDP. We analyze the di#culties inherent in current methods and present robust algorithms.
Computing the Global Optimum . . .
, 2008
"... 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 bo ..."
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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, xℓdxℓf  → 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.
Preprocessing and . . .
, 2013
"... This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming r ..."
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This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The
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, 2012
"... backwards stability. AMS subject classifications: 90C46, 90C22, 90C25, 49K40, 65K10 This paper presentsa backwardstable preprocessingtechnique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible ..."
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backwards stability. AMS subject classifications: 90C46, 90C22, 90C25, 49K40, 65K10 This paper presentsa backwardstable preprocessingtechnique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The
Factors that impact solution run times of arcbased formulations of the Vehicle Routing Problem ∗
, 2005
"... It is well known that the Vehicle Routing Problem (VRP) becomes more difficult to solve as the problem size increases. However little is known about what makes a VRP difficult or easy to solve for problems of the same size. In this paper we investigate the effect of the formulation and data paramete ..."
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It is well known that the Vehicle Routing Problem (VRP) becomes more difficult to solve as the problem size increases. However little is known about what makes a VRP difficult or easy to solve for problems of the same size. In this paper we investigate the effect of the formulation and data parameters on the efficiency with which we can obtain exact solutions to the VRP with a general IP solver. Our results show that solution run times for arcbased formulations with exponentially many constraints are mostly insensitive to problem parameters, whereas polynomial arcbased formulations, which can solve larger problems because of the smaller memory requirement, are sensitive to problem parameters. For instance, we observe that solution times for polynomial formulations significantly decrease for larger capacities and number of vehicles.