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This linear programming relaxation allows a fraction of each item. With positive weights, sort decreasing profit-to-weight ratios and fill capacity in that order, using at most one fractional item. An exchange argument proves optimality. Negative-profit items are omitted and positive-profit zero-weight items are included for free.

If a vector $h$ gives the strict separation, a common point would satisfy $h^{T}x<h^{T}x$, which is impossible.

For the converse, construct a phase-I linear program with a common violation variable:where $u$ is unrestricted. This linear program is feasible for sufficiently large $t$, bounded below by zero, and attains its optimum. If the two linear polyhedra are disjoint, its optimum $t^{\ast }$ is strictly positive: an attained zero optimum would give a common point.

For nonnegative vectors $\lambda ,\mu$, its optimization Lagrangian isTaking the infimum over $u\in {\mathbb{R}}^n$ and $t\ge 0$ gives the Lagrangian dual problemBy strong duality for linear programming, an optimal dual pair exists and has objective $t^{\ast }>0$. It supplies the infeasibility certificateTake $h=A^T\lambda =-C^T\mu$. Every $x\in P$ and $y\in Q$ obeysThe assumed nonemptiness of each linear polyhedron also ensures $h\ne 0$: if $h=0$, their feasibility would force both ${\lambda }^{T}b$ and ${\mu }^{T}d$ to be nonnegative. This proves the strict separation of disjoint linear polyhedra using Lagrangian duality.

A semidefinite program optimizes a linear function subject to equalities between affine functions and positive semidefinite matrix inequalities. It generalizes linear programming: a diagonal matrix is a positive semidefinite matrix exactly when its diagonal entries are nonnegative. Matrix trace expresses the objective as $⟨C,X⟩=\mathrm{tr}(CX)$ for real symmetric matrices.

For standard-form linear programming constraints $Ax=b$ of full row rank, a simplex basis is a set of columns forming an invertible square matrix $B$. The remaining variables are nonbasic; setting them to zero gives the basic solution $x_B=B^{-1}b$. It is feasible if these basic variables are nonnegative. A pivot exchanges one basic and one nonbasic column while maintaining this representation.