= Strict feasibility for conic duality
For a <proper cone> $K$, suppose a point satisfies $Ax-b\in\operatorname{int}K$ and $p_*$ is finite. This generalized <Slater condition> gives <strong duality> and attainment of the <conic dual problem>'s optimum. It does not by itself assert primal attainment.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§5.9.1, printed p. 265: generalized Slater condition, strong duality and dual attainment]{ref}
The two worked examples have strict primal points: $(u,v)=(2,2)$ has slack $(2,2,1)$ in the interior of the <nonnegative orthant>, and $(t,z)=(2,0)$ has slack $(2,1,0)$ in the interior of the <second-order cone>. Both optima were already certified by explicit feasible pairs. <Slater's condition> explains why a matching optimal <dual variable> can be expected under these hypotheses; checking a supplied zero-gap pair needs only <weak duality>.
Changing cone geometry preserves the bound mechanism. For a <positive semidefinite cone>, the <dual cone> pairing is the <Frobenius inner product> of two <matrices>. This connects the same certificate idea to <semidefinite programming>, while the <central path> uses strictly interior primal and dual points to approach vanishing gap.
In 1995, <Lieven Vandenberghe> and <Stephen Boyd> presented a primal-dual <interior-point method> for <convex optimization> with <matrix> inequalities. Their algorithmic study included exploiting problem structure and allowing incomplete <Conjugate gradient method> solves within an iteration. This illustrates how the primal-dual viewpoint guides computation as well as checking a final answer.<A Primal-Dual Potential Reduction Method for Problems Involving Matrix Inequalities by Lieven Vandenberghe and Stephen Boyd>[Publication record and authored abstract: A Primal-Dual Potential Reduction Method for Problems Involving Matrix Inequalities, Mathematical Programming, Series B, 69(1), 205–236 (1995)]{ref}
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