In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular we define a general … Read more
We explicitly calculate universal barrier functions for cones generated by (weakly) Chebyshev systems over finite sets. We show that universal barrier functions corresponding to Chebyshev systems on intervals are obtained as limits of universal barrier functions of their discretizations. The results are heavily rely upon classical work of M. Krein, A. Nudelman and I.J. Schoenberg … Read more
We propose an algorithm based on Barvinok’s counting algorithm for P -> max {c’x | Ax
This paper presents a strategy for choosing the initial point, slacks and multipliers in interior methods for nonlinear programming. It consists of first computing a Newton-like step to estimate the magnitude of these three variables and then shifting the slacks and multipliers so that they are sufficiently positive. The new strategy has the option of … Read more
Virtual private networks (VPN) are often used to distribute live content, such as video or audio streams, from a single source to a large number of destinations. Streaming caches or splitters are deployed in these multicast networks to allow content distribution without overloading the network. In this paper, we consider two combinatorial optimization problems that … Read more
The standard quadratic program (QPS) is $\min_{x\in\Delta} x’Qx$, where $\Delta\subset\Re^n$ is the simplex $\Delta=\{ x\ge 0 : \sum_{i=1}^n x_i=1 \}$. QPS can be used to formulate combinatorial problems such as the maximum stable set problem, and also arises in global optimization algorithms for general quadratic programming when the search space is partitioned using simplices. One … Read more
In this paper we study a special class of convex optimization problems called {\em conically ordered convex programs}\/ (COCP), where the feasible region is given as the level set of a vector-valued nonlinear mapping, expressed as a nonnegative combination of convex functions. The nonnegativity of the vectors is defined using a pre-described conic ordering. The … Read more
Primal-Dual Interior-Point Methods (IPMs) have shown their power in solving large classes of optimization problems. In this paper a self-regular proximity based Infeasible Interior Point Method (IIPM) is proposed for linear optimization problems. First we mention some interesting perties of a specific self-regular proximity function, studied recently by Peng and Terlaky, and use it to … Read more
It is well known that the so-called first-order predictor-corrector methods working in a large neighborhood of the central path are among the most efficient interior-point methods (IPMs) for linear optimization (LO) problems. However, the best known iteration complexity of this type of methods is $O\br{n \log\frac{(x^0)^Ts^0}{\varepsilon}}$. It is of interests to investigate whether the complexity … Read more
It is known that linear conic systems are a special case of set-valued sublinear mappings. Hence the latter subsumes the former. In this note we observe that linear conic systems also contain set-valued sublinear mappings as a special case. Consequently, the former also subsumes the latter. CitationOperations Research Letters 32 (2004) pp. 463–467.