Global Maximization of a Generalized Concave Multiplicative Function

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown; computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.     

一类特殊线性双层规划的对偶规划

本文首先给出值型线性双层规划的等价形式 ,然后讨论了非增的值型线性双层规划的 Johri一般对偶规划 ,并且说明了其对偶间隙等于零 ,最后说明了它们最优解的关系     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques

In a recent work, we introduced the concept of convex extensions for lower semi-continuous functions and studied their properties. In this work, we present new techniques for constructing convex and concave envelopes of nonlinear functions using the theory of convex extensions. In particular, we develop the convex envelope and concave envelope of z=x/y over a hypercube. We show that the convex envelope is strictly tighter than previously known convex underestimators of x/y. We then propose a new relaxation technique for fractional programs which includes the derived envelopes. The resulting relaxation is shown to be a semidefinite program. Finally, we derive the convex envelope for a class of functions of the type f(x,y) over a hypercube under the assumption that f is concave in x and convex in y.     

Formulations for a problem of petroleum transportation

Oil tankers play a fundamental role in every offshore petroleum supply chain and due to its high price, it is essential to optimize its use. Since this optimization requires handling detailed operational aspects, complete optimization models are typically intractable. Thus, a usual approach is to solve a tactical level model prior to optimize the operational details. In this case, it is desirable that tactical models are as precise as possible to avoid too severe adjustments in the next optimization level. In this paper, we study tactical models for a crude oil transportation problem by tankers. We did our work on the top of a previous paper found in the literature. The previous model considers inventory capacities and discrete lot sizes to be transported, aiming to meet given demands over a finite time horizon. We compare several formulations for this model using 50 instances from the literature and proposing 25 new harder ones. A column generation-based heuristic is also proposed to find good feasible solutions with less computational burden than the heuristics of the commercial solver used.     

ALTERNATELY LINEARIZED IMPLICIT ITERATION METHODS FOR SOLVING QUADRATIC MATRIX EQUATIONS

A numerical solution of the quadratic matrix equations associated with a nonsingular M-matrix by using the alternately linearized implicit iteration method is considered. An iteration method for computing a nonsingular M-matrix solution of the quadratic matrix equations is developed, and its corresponding theory is given. Some numerical examples are provided to show the efficiency of the new method.     

Two stage linear programming under uncertainty with 0–1 integer first stage variables

Stochastic programs with continuous variables are often solved using a cutting plane method similar to Benders' partitioning algorithm. However, mixed 0–1 integer programs are also solved using a similar procedure along with enumeration. This similarity is exploited in this paper to solve two stage linear programs under uncertainty where the first stage variables are 0–1. Such problems often arise in capital investment. A network investment application is given which includes as a special case a coal transportation problem.     

Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion

In this paper, we derive explicit bounds for the Kolmogorov distance in the CLT and we prove the almost sure CLT for the quadratic variation of the sub-fractional Brownian motion. We use recent results on the Stein method combined with the Malliavin calculus and an almost sure CLT for multiple integrals.     

Hanson's duality theorem in nonsmooth programming

A nonlinear program with inequality and equality constraints, generated by lipschitzian functions in a real Banach space is considered. The sufficiency of the Kuhn-Tucker optimality conditions at a point is established, using the function subdifferentials which generate the program. Also. in nonsmooth frame, Hanson's converse duality theorem from the convex programming is generalized     

Solving nonlinear equations by a new derivative free iterative method

We develop a new simple iteration formula, which does not require any derivatives of f(x), for solving a nonlinear equation f(x) = 0. It is proved that the convergence order of the new method is quadratic. Furthermore, the new method can approximate complex roots. By several numerical examples we show that the presented method will give desirable approximation to the root without a particularly good initial approximation and be efficient for all cases, regardless of the behavior of f(x).