<p>We consider an <i>n</i>-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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10107_2025_2212_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{n}\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>n</mtext> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, is given by a conic inequality, and present the lower envelope for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10107_2025_2212_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm {n=2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10107_2025_2212_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm {n=2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Convex envelopes of bounded monomials on two-variable cones

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2025年3月17日

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Pietro Belotti

摘要

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 $\textrm{n}\ge 2$ $n \geq 2$ , is given by a conic inequality, and present the lower envelope for $\mathrm {n=2}$ $n = 2$ . 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 $\mathrm {n=2}$ $n = 2$ .