The interval-Newton approach provides the power to solve nonlinear equation solving and global optimization problems with complete mathematical and computational certainty. The primary drawback to this approach is that computation time requirements may become quite high. In this paper, a strategy for using linear programming (LP) techniques to improve computational efficiency is considered. In ...

In 1991 Lorentzen and Nilsen showed how to use linear programming to prove lower bounds on the size of difference triangle sets. In this note we show how to improve these bounds by including additional valid linear inequalities in the LP formulation. We also give some new optimal difference triangle sets found by computer search. AMS Subject Classification: 05B10 Following Kløve [2] we define a...

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined sparse component analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Cont...

We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different interior-point algorithms. We obtain an improved comple...

For the linear programming (LP) problems, perhaps infeasible, with block-angular matrices of constraints, two parallel second order optimization methods are proposed. Being applied to initial LP problem they give its solution if this problem is solvable, and automatically deliver a solution of some relaxed LP problem, otherwise. The methods use penalty and barrier functions as well as Tikhonov ...

We demonstrate that a conic quadratic problem min x { ex ∣Ax ≥ b, ‖A`x− b`‖2 ≤ c` x− d`, ` = 1, ...,m } , ‖y‖2 = √ yT y, (CQP) is “polynomially reducible” to Linear Programming. We demonstrate this by constructing, for every ∈ (0, 12 ], an LP program (explicitly given in terms of and the data of (CQP)) min x,u { ex ∣P ( x u ) + p ≥ 0 } (LP) with the following properties: (i) the number dim x+ d...

In this section we consider another method for solving the set cover problem approximately. The method uses randomized rounding of the solution obtained from the linear programming (LP) relaxation of set cover. The basic idea behind this algorithm is first to solve the LP relaxation for the set cover problem. We can think of the solution ~x as a probability to either select a set or not. Note t...

While (LP1) belongs to the complexity class P, (IP1) is an NP-hard problem. The most frequently used algorithm to solve LP problems, the simplex method, is efficient in practice but has an exponential worst case running time. A detailed discussion and a list of useful pointers in the literature regarding the complexity of LP and IP problems can be found in the book of Schrijver [72]. This secti...

Linear programming (LP) relaxations are a popular method to attempt to find a most likely configuration of a discrete graphical model. If a solution to the relaxed problem is obtained at an integral vertex then the solution is guaranteed to be exact and we say that the relaxation is tight. We consider binary pairwise models and introduce new methods which allow us to demonstrate refined conditi...

The introduction of a standard set of linear programming problems, to be found in NETLIB/LP/DATA, had an important impact on measuring, comparing and reporting the performance of LP solvers. Until recently the efficiency of new algorithmic developments has been measured using this important reference set. Presently, we are witnessing an ever growing interest in the area of quadratic programming...