摘要
研究了一类具有超矩形约束的特殊三次规划问题,利用目标函数的三次上估计函数与下估计函数推导出该问题的全局最优必要性与充分性条件。首先,构造如下形式的三次上估计函数与下估计函数h(x)=l(x)-l(x)+f(x),其中f(x)是目标函数,l(x)=∑n i=113αix3i+12xTQx+(b+(A-Q)x)Tx。接着利用三次上估计函数建立判断一个可行点是全局最优点的全局最优必要性条件。然后利用三次下估计函数建立判断一个可行点是全局最优点的全局最优充分性条件:τipi(xi)+τimin{γipi(ui),γipi(vi)}≥0,i∈I,τipi(xi)≤0与pi(xi)=0,i∈J。一些实例说明了这些全局最优必要性与充分性条件的有效性与可行性。
In this paper,global optimality necessary and sufficient conditions are presented for a class cubic programming problems involving rectangle constraints via cubic overestimators and underestimators. Firstly,we construct cubic overestimators and underestimators of objective function h( x) = l( x)-l( x) + f( x),where f( x) is objective function and l( x) = ∑n i = 113αix3i+12xTQx +( b +( A-Q) x)Tx. Secondly,by utilizing cubic overestimators,we present some necessary global optimality conditions for a feasible point to be a global minimizer of cubic progamming problems involving rectangle constraints. Then,the following sufficient conditions are established for a feasible point to be a global minimizer of cubic progamming problems using cubic underestimators:τipi(xi) +τimin{ γipi( ui),γipi( vi) } ≥0,i∈I,τ^ipi( x^i) ≤0,pi( xi) =0,i∈J. Finally,some examples are used to illustrate effectiveness and feasibility of global optimality conditions.
引文
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