具有超矩形约束的三次规划的全局最优性条件
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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.
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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[2]Nesterov Y.Accelerating the cubic regularization of Newtons method on convex problems[J].Math Program,2008,112(1):159-181.
[3]Lin C S,Chang P R,Luh J Y S.Formulation and optimization of cubic polynomial joint trajectories for industrial robots[J].IEEE Transact Autom Control AC,1983,28(12):1066-1074.
[4]祁云峰,吴至友.混合整数二次规划的全局充分性最优条件[J].重庆师范大学学报:自然科学版,2010,27(5):1-4.Qi Y F,Wu Z Y.Sufficient global optimality conditions for mixed-integer quadratic minimization problem with inequality constraints[J].Journal of Chongqing Normal University:Natural Science,2010,27(5):1-4.
[5]王杉林.一类非凸二次规划问题的全局最优性充分条件[J].重庆师范大学学报:自然科学版,2008,25(4):5-7.Wang S L.Global optimality conditions of non-convex quadratic problem with quadratic equality constraints[J].Journal of Chongqing Normal Universit:Natural Science.2008,25(4):5-7.
[6]张甲,田志远,李敬玉.一类非凸二次规划问题的全局最优性条件[J].青岛大学学报:自然科学版,2010,23(3):20-23.Zhang J,Tian Z Y,Li J Y.Sufficient global optimality conditions for some nonconvex quadratic program problems[J].Journal of Qingdao University:Natural Science Edition,2010,23(3):20-23.
[7]WangY J,Liang Z.Global optimality conditions for cubic minimization problem with box or binary constraints[J].Journal of Global Optimization,2010,47(4):583-595.
[8]Wu Z Y,Rubinov A M.Global optimality conditions for some classes of optimization problems[J].Journal of Mathematical Analysis and Applications,2010,145(1):164-185.
[9]Zhou X G,Cao B Y.New global optimality conditions for cubic minimization problem with box or bivalent constraints[J].Pacific Journal of Optimization,2012,8(3):631-647.
[10]Dahl G.A note on diagonally dominant matrices[J].Linear Algebra Appl,2000,317(3):217-224.