Convex envelopes of bounded monomials on two-variable cones
研究了在上下界和两个齐次线性不等式约束下的n变量单项式函数，给出了n≥2时上包络的锥不等式形式，以及n=2时的下包络和凸包体积，对混合整数非线性优化问题的求解有参考价值。
Abstract We consider an n -variate monomial function that is restricted both in value by lower and upper bounds and in domain by two homogeneous linear inequalities. Monomial functions are building blocks for the class of Mixed Integer Nonlinear Optimization problems, which has many practical applications. We show that the upper envelope of the function in the given domain, for $$\textrm{n}\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>n</mml:mtext> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , is given by a conic inequality, and present the lower envelope for $$\mathrm {n=2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We also discuss branching rules that maintain these convex envelopes and their applicability in a branch-and-bound framework, then derive the volume of the convex hull for $$\mathrm {n=2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .