English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The trust-region problem, which minimizes a nonconvex quadratic function overa ball, is a key subproblem in trust-region methods for solving nonlinearoptimization problems. It enjoys many attractive properties such as an exactsemi-definite linear programming relaxation (SDP relaxation) and strongduality. Unfortunately, such properties do not, in general, hold for anextended trust-region problem having extra linear constraints. This paper showsthat two useful and powerful features of the classical trust-region problemcontinue to hold for an extended trust-region problem with linear inequalityconstraints under a new dimension condition. First, we establish that the classof extended trust-region problems has an exact SDP-relaxation, which holdswithout the Slater constraint qualification. This is achieved by proving that asystem of quadratic and affine functions involved in the model satisfies arange-convexity whenever the dimension condition is fulfilled. Second, we showthat the dimension condition together with the Slater condition ensures that aset of combined first and second-order Lagrange multiplier conditions isnecessary and sufficient for global optimality of the extended trust-regionproblem and consequently for strong duality. Finally, we show that thedimension condition is easily satisfied for the extended trust-region modelthat arises from the reformulation of a robust least squares problem (LSP) aswell as a robust second order cone programming model problem (SOCP) as anequivalent semi-definite linear programming problem. This leads us to concludethat, under mild assumptions, solving a robust (LSP) or (SOCP) undermatrix-norm uncertainty or polyhedral uncertainty is equivalent to solving aSDP and so, their solutions can be validated in polynomial time.

Trust--Region Problems with Linear Inequality Constraints: Exact SDP Relaxation, Global Optimality and Robust Optimization