We investigate in this paper the duality gap between quadratic knapsack problem and its Lagrangian dual or semidefinite programming relaxation. We characterize the duality gap by a distance measure from set {0, 1}<Superscript> n </Superscript> to certain polyhedral set and demonstrate that the duality gap can be reduced...</superscript>

This article presents an analysis of the convergence order of Taylor models and McCormick-Taylor models, namely Taylor models with McCormick relaxations as the remainder bounder, for factorable functions. Building upon the analysis of McCormick relaxations by Bompadre and Mitsos (J Glob Optim...

In this work, by using weak conjugate maps given in (Azimov and Gasimov, in Int J Appl Math 1:171–192, <CitationRef CitationID="CR1">1999</CitationRef>), weak Fenchel conjugate dual problem, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${(D_F^w)}$$</EquationSource> </InlineEquation> , and weak Fenchel Lagrange conjugate dual problem <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${(D_{FL}^w)}$$</EquationSource> </InlineEquation> are constructed. Necessary and sufficient conditions for...</equationsource></inlineequation></equationsource></inlineequation></citationref>

This paper investigate a class of semi-supervised support vector machines (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\text{ S }^3\mathrm{VMs}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mspace width="4.pt"/> <mtext>S</mtext> <msup> <mspace width="4.pt"/> <mn>3</mn> </msup> <mi mathvariant="normal">VMs</mi> </mrow> </math> </EquationSource> </InlineEquation>) with arbitrary norm. A general framework for the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\text{ S }^3\mathrm{VMs}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mspace width="4.pt"/> <mtext>S</mtext> <msup> <mspace width="4.pt"/> <mn>3</mn> </msup> <mi mathvariant="normal">VMs</mi> </mrow> </math> </EquationSource> </InlineEquation> was first constructed based on a robust DC (Difference of Convex...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>

We present a new way to solve generalized Nash equilibrium problems. We assume the feasible set to be compact. Furthermore all functions are assumed to be polynomials. However we do not impose convexity on either the utility functions or the action sets. The key idea is to use Putinar’s...

We propose and study a new method, called the Interior Epigraph Directions (IED) method, for solving constrained nonsmooth and nonconvex optimization. The IED method considers the dual problem induced by a generalized augmented Lagrangian duality scheme, and obtains the primal solution by...

We consider the optimal operation of a hydroelectric plant supplemented by a set of thermal plants. The initial model gives rise to a discrete minimization problem with a convex cost function, submitted to both concave and convex restrictions. The geometry of the water reservoir is taken into...

We present a new way to solve generalized Nash equilibrium problems. We assume the feasible set to be compact. Furthermore all functions are assumed to be polynomials. However we do not impose convexity on either the utility functions or the action sets. The key idea is to use Putinar’s...