拟凹函数

Properties of efficient portfolios and the efficient frontier to the model are systematically analyzed. Our main results concerning the properties are: every efficient portfolio can be solved by minimizing portfolio risk under a given level of portfolio return or by maximizing portfolio return under a given level of portfolio risk; on the efficient frontier, the risk is a convex and strictly increasing function of the return and the return is a concave and strictly increasing function of the risk; the utility function on the efficient frontier can be expressed as a quasi-concave function of the risk or the return if the investor's utility function is quasi-concave.

从理论上系统地对该模型下的有效投资组合和有效前沿的性质进行了分析，结果表明：每一个有效投资组合可通过在给定期望收益水平的条件下最小化投资组合风险来获得，或者在给定风险水平的条件下最大化期望投资组合收益来获得；在有效前沿上，风险是收益的严格单调递增凸函数，收益是风险的严格单调递增凹函数；当投资者的效用函数是拟凹函数时，则有效前沿上的效用可表达成风险或收益的拟凹函数。

Thirdly,quasi-normal inequalities of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales areidentified by the use of martingale transforms and by the construction of convex or concave function method;on this basis and with the help of previsiblity or regu-larity,Burkholder-Davis-Gundy\'s inequality of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales are iden-tified by the application of Hardy-Lorentz interpolation theory.At the same time,bythe use of G.

再次，应用鞅变换和构造凸或凹函数方法证明了标量值可料树鞅的α-方极大算子和α-条件方极大算子的拟范数不等式；然后，在这些拟范数不等式的基础上，应用Hardy-Lorentz空间插值方法证明了当树鞅是可控或正规树鞅时关于标量值可料树鞅α-方极大算子和α-条件方极大算子的Burkholder-Davis-Gundy's不等式成立。

Then, using the interval arithmetic tool, construction methods of several quasi convex-concave extension are presented for general functions which are provided with arithmetic expression.

其次，对由算术表达式给定的一般函数，利用区间算术工具给出了几种拟凸-凹扩张的构造方法。

Firstly, a new quasi convex-concave extension definition is introduced to study general constrained optimization problem, which is much approximate function's shape than the interval extension.

首先，对一般约束优化问题，引入一新的概念，即拟凸-凹扩张，它比区间扩张能更好地逼近函数的形状。

Quasi-convex relaxation algorithm, which aims to solve ordinary constraint global optimization problems, is produced by a combination of branch-and-bound method with quasi-convex lower bound and quasi-concave upper bound functions developed from quasi convex-concave extension. This algorithm extends the interval algorithm with box constraint to general constraint global optimization problem.

利用拟凸-凹扩张中构造出的函数的拟凸下界函数和拟凹上界函数，与分枝定界方法相结合，建立了求一般约束全局优化问题的拟凸松弛算法，这一算法是将盒子约束的区间算法推广到一般约束优化问题中。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。