We derive the duality relation for the Hilbert series H (d; z) of almost symmetric numerical semigroup S (d) combining it with its dual H ( d; z ) . On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps Fj , generators di and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd i...

We present new results for the Frank-Wolfe method (also known as the conditional gradient method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple averaging and constant step-sizes. We also develop step-size rules and computational guarantees that depend naturally on the warm-start quality of the initial (...

In many network routing problems several conflicting objectives must be considered. Even for the bi-objective shortest path problem, generating and presenting the whole set of nondominated solutions (paths) to a decision maker, in general, is not effective because the number of these paths can be very large. Interactive procedures are adequate to overcome these drawbacks. Current et al. [1] pro...

We present new results for the conditional gradient method (also known as the Frank-Wolfe method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple averaging and constant step-sizes. We also develop step-size rules and computational guarantees that depend naturally on the warm-start quality of the initial (...

Digitally-disadvantaged languages face multiple inequities in the digital sphere, with their speaker communities frequently experiencing duality of neglect and surveillance. These suffer from gaps support, when support does exist, it often makes vulnerable to surveillance content moderation.

In 1951, Dantzig [3] showed the equivalence of linear programming and two-person zero-sum games. However, in the description of his reduction from linear programming to zero-sum games, he noted that there was one case in which his reduction does not work. This also led to incomplete proofs of the relationship between the Minmax Theorem of game theory and the Strong Duality Theorem of linear pro...

In 1951, Dantzig showed the equivalence of linear programming problems and two-person zerosum games. However, in the description of his reduction from linear programs to zero-sum games, he noted that there was one case in which the reduction does not work. This also led to incomplete proofs of the relationship between the Minimax Theorem of game theory and the Strong Duality Theorem of linear p...

In solving certain optimization problems, the corresponding Lagrangian dual problem is sometimes solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that for all optimization problems the optimal dual function value is smaller than or equal to...