The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.

1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x

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In general, results in the area of developing efficient algorithms for solving large-scale optimization problems will be of great importance in improving the efficiency of manufacturing systems, communication networks, aircraft routing, multi-commodity-flow operations, and resources planning. Strengthening research in this area will definitely contribute to the national interest in industrial competitiveness and scientific knowledge.

Path-following linear programming (LP) algorithms generate a sequence of points within certain neighborhoods of a central-path C, which prevent iterates from prematurely getting too close to the boundary of the feasible region. Depending on their norm used, these neighborhoods include N2(fi), N1(fi) and N \Gamma 1(fi), where fi 2 (0; 1), and C ae N2(fi) ae N1(fi) ae N \Gamma 1(fi) for each fi 2 (0; 1): A paradox is that among all existing (infeasible or feasible) path-following algorithms, the theoretical iteration complexity, O(pnL), of small-neighborhood (N2) algorithms is significantly better than the complexity, O(nL), of wide-neighborhood (N \Gamma 1) algorithms, while in practice wide-neighborhood algorithms outperform small-neighborhood ones by a big margin. Here, n is the number of LP variables and L is the LP data length. In this paper, we present an O(n n+1 2n L)-iteration (infeasible) primal-dual high-order algorithm that uses wide neighborhoods. Note that this iteration bound is asymptotical O(pnL), i.e., the best bound for small-neighborhood algorithms, as n increases.

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