Determining the set of all optimal solutions of a linear program with interval data is one of the main problems discussed in interval optimization. We review two methods based on duality in linear programming, which are used to approximate the optimal set. Additionally, another decomposition method based on complementary slackness is proposed. This method provides the exact description of the o...

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researche...

We describe an O(nh min{log U, n log n}) capacity scaling algorithm for the minimum cost submodular flow problem. Our algorithm modifies and extends the Edmonds–Karp capacity scaling algorithm for minimum cost flow to solve the minimum cost submodular flow problem. The modification entails scaling a relaxation parameter δ. Capacities are relaxed by attaching a complete directed graph with unifo...

The clinical and radiological features in seven patients who had asymmetric muscular atrophy of the hand and forearm when young are reported and a new hypothesis for its aetiology is proposed. Investigation of body growth curves (a surrogate for velocity of arm growth) showed close relation between (a) the age when the body height increased most rapidly and the onset age of this disorder, and (...

Supplementary Material 6 Auxiliary Lemmas: Proof of Lemma 3 Proof. We can rewrite (6) as an optimization problem over the ℓ 1 /ℓ 2 ball of radius C for some C(λ n) < ∞. 1,2 = C for all optimal primal solution Θ \r. By definition of the ℓ 1 /ℓ 2 subdifferential, we know that for any column u ∈ V \{r}, we have (ˆ Z \r) u 2 ≤ 1. Considering the necessary optimality condition ∇ℓ (ˆ Θ \r) + λ n ˆ Z ...

There are similarities between semidefinite programming and linear programming in theory and practice, e.g., duality theory (see Bellman and Fan [5]), the role of complementary slackness, and efficient solution techniques using interior-point methods (see Nesterov and Nemirovski [11], Wright, [20]). Semidefinite programming has been developed both theoretically and practically for the past few ...

Consider a proper cone K ⊂ < and its dual cone K. It is well known that the complementary slackness condition xs = 0 defines an n-dimensional manifold C(K) = { (x, s) : x ∈ K, s ∈ K, xs = 0 } ⊂ <×<. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. This fact proves to be very useful when optimizing over such cones, therefore it is natural to look for ...

We present a branch-and-cut algorithm for solving discrete bilevel linear programs where the upper-level variables are binary and the lower-level variables are either pure integer or pure binary. This algorithm performs local search to find improved bilevelfeasible solutions. We strengthen the relaxed node subproblems in the branch-and-cut search tree by generating cuts to eliminate all of the ...

This paper describes Patent Number U.S. 8,046,316 B2, titled “Cyclic Combinatorial Method and System”, issued by the US Patents and Trademarks Office on October 25, 2011. The patent is based on a combinatorial algorithm to solve cyclic optimization problems. First, the algorithm identifies cyclically distinct solutions of such problems by enumerating cyclically distinct combinations of the basi...