Perturbed Farkas-Type Lemmas in Conic Optimization

• Perturbed Farkas-type lemmas are algebraic extensions that characterize feasibility, zero duality gap, and strong duality in conic optimization.
• They incorporate cost-side and constraint-side perturbations using graphical and epigraphical sets to capture value function changes.
• The framework replaces traditional topological conditions with algebraic slice equalities, unifying classical convex analysis and infinite-dimensional settings.

Perturbed Farkas-type lemmas embody algebraic formulations of the classical Farkas lemma, extended to infinite-dimensional conic linear programming over dual pairs of vector spaces, and enriched by cost-side or constraint-side perturbations. These results enable comprehensive characterizations of feasibility, zero duality gap, and strong duality for conic optimization problems, building upon the theory of convex cones, dual pairs, and adjoint operators. The underlying framework operates purely algebraically, broadening the scope beyond classical topological assumptions, and leverages the convex-analytic separation principle via “graphical” and “epigraphical” construction (Khanh et al., 15 Jan 2026).

1. Dual Pairs, Conic Linear Programs, and Notation

The development of perturbed Farkas-type lemmas proceeds in the setting of separated dual pairs of real vector spaces $(X,Y)$, $(Z,W)$, for which each space serves as the algebraic dual of the other under a bilinear pairing $\langle\cdot,\cdot\rangle$, ensuring separation: $\langle x, y \rangle = 0$ $\forall y \implies x = 0$ and vice versa. Given a linear map $A: X \to Z$, its adjoint $A^*: W \to Y$ is defined to preserve dual pairings: $$\langle x, A^*w \rangle = \langle Ax, w \rangle \quad \forall x \in X,\, w \in W.$$

Conic linear optimization utilizes closed convex cones $P \subset X$, $Q \subset Z$, with respective positive dual cones

$$P^* = \{ y \in Y : \langle x, y \rangle \geq 0 \,\, \forall x \in P \}, \qquad Q^* = \{ w \in W : \langle z, w \rangle \geq 0 \,\, \forall z \in Q \}.$$

The primal and dual conic programs take the form: $$(\mathcal{P}) \quad \min_{x \in P} \{ \langle x, c \rangle : Ax - b \in Q \},\qquad (\mathcal{D}) \quad \max_{w \in Q^*} \{ \langle b, w \rangle : -A^*w + c \in P^* \},$$ with feasible regions $F(\mathcal{P})$, $F(\mathcal{D})$, and optimal values $$\mathrm{val}(\mathcal{P}), \mathrm{val}(\mathcal{D})$$ governed by weak duality ($\geq$). Strong duality, or “zero duality gap”, occurs if $$\mathrm{val}(\mathcal{P}) = \mathrm{val}(\mathcal{D})$$.

2. Cost Perturbations, Value Functions, and Graphical Sets

Perturbed Farkas lemmas hinge on value functions reflecting perturbations to objective or constraint data:

• Dual-side perturbation ($z \in Z$):

$$v_{\mathcal{D}}(z) := \sup \{ \langle b + z, w \rangle : w \in F(\mathcal{D}) \}.$$

• Primal-side perturbation ($y \in Y$):

$$v_{\mathcal{P}}(y) := \inf \{ \langle x, c - y \rangle : x \in F(\mathcal{P}) \}.$$

This induces:

• Epigraphical set ($\mathcal{N}$): The region above $v_{\mathcal{D}}(z)$ in $Z \times \mathbb{R}$,

$$\mathcal{N} := \{ (z, r) : r \geq v_{\mathcal{D}}(z) \}.$$

• Hypergraphical set ($\mathcal{M}$): The region below $v_{\mathcal{P}}(y)$ in $Y \times \mathbb{R}$,

$$\mathcal{M} := \{ (y, r) : r \leq v_{\mathcal{P}}(y) \}.$$

Auxiliary sets $\mathcal{H}$ and $\mathcal{K}$ correspond to unperturbed “shifted” formulations, always satisfying $\mathcal{H} \subset \mathcal{N}$ and $\mathcal{K} \subset \mathcal{M}$, with primal and dual optimal values retrievable as infima or suprema on these sets evaluated at the zero slices.

3. Convex Separation Characterizations

A key feature of the theory is the convex separation principle, classifying infeasibility in graphical sets via supporting affine functionals:

• For $\mathcal{N}$: $(z, r) \notin \mathcal{N}$ iff there exists $(w, \sigma) \in W \times \mathbb{R}$, $\alpha > 0$ such that for every $(z', r') \in \mathcal{H}$,

$$\langle z', w \rangle + \sigma r' \leq \langle z, w \rangle + \sigma r - \alpha.$$

• For $\mathcal{M}$: $(y, r) \notin \mathcal{M}$ iff separated by $(x, \tau) \in X \times \mathbb{R}$, $\beta > 0$ in the analogous fashion.

These results generalize Kretschmer’s classical closedness conditions for conic programs, replacing topological closure requirements with slice equalities between the auxiliary and graphical sets (i.e., $\mathcal{H} = \mathcal{N}$, $\mathcal{K} = \mathcal{M}$).

4. Perturbed and Unperturbed Farkas-Type Lemmas

The central content is embodied in four Farkas-type lemmas, each relating a dual-type condition to a primal-type condition for every slice in the space of perturbations.

Perturbed Farkas Lemma 1:

Fix $z \in Z$, the following equivalence holds without extra assumptions:

• Slice equality: $$\mathcal{H} \cap (\{z\} \times \mathbb{R}) = \mathcal{N} \cap (\{z\} \times \mathbb{R})$$.
• Dual-primal equivalence: For all $\rho \in \mathbb{R}$,
  • (a) $\langle b + z, w \rangle \leq \rho$ for all $w \in Q^*$, $-A^*w + c \in P^*$,
  • (b) $\exists\, x \in P$ with $Ax - b - z \in Q$ and $\langle x, c \rangle \leq \rho$.

Perturbed Farkas Lemma 2:

Fix $y \in Y$, equivalently for all $\lambda \in \mathbb{R}$,

• (a) $\langle x, c - y \rangle \geq \lambda$ for all $x \in P$, $Ax - b \in Q$,
• (b) $\exists\, w \in Q^*$ with $-A^*w + c - y \in P^*$ and $\langle b, w \rangle \geq \lambda$.

The unperturbed versions result by restricting the perturbation to zero ($z = 0$ or $y = 0$), recovering the classical Farkas lemma for conic systems.

5. Strong Duality and Zero Duality Gap via Farkas Lemmas

Perturbed Farkas-type lemmas yield strong duality formulations:

• For fixed $z$:

$$\min_{x \in P,\ Ax - b - z \in Q} \langle x, c \rangle = \sup_{w \in Q^*,\ -A^*w + c \in P^*} \langle b + z, w \rangle$$

iff the slice identity $$\mathcal{H} \cap (\{z\} \times \mathbb{R}) = \mathcal{N} \cap (\{z\} \times \mathbb{R})$$ holds and the slice is nonempty.

• For fixed $y$: analogous statement for primal-side perturbation.

Zero duality gap in the unperturbed optimization problem ($z = 0, y = 0$) emerges as the base-slice identity: $$\mathcal{H} \cap (\{0\} \times \mathbb{R}) = \mathcal{N} \cap (\{0\} \times \mathbb{R})$$, and the analogous statement for $\mathcal{K}$ and $\mathcal{M}$ (Khanh et al., 15 Jan 2026).

6. Sufficient Algebraic and Topological Conditions

Algebraic criteria guaranteeing equalities $\mathcal{H} = \mathcal{N}$ and $\mathcal{K} = \mathcal{M}$ include:

• Existence of feasible points in the dual (resp. primal) and no bidual gap, i.e., $(P^*)^* = i_X(P)$.
• Dually, nonemptiness of primal feasibility yields the corresponding statement for the hypergraphical sets.

Topological characterizations assert that, with the weak topology $\sigma(Z \times \mathbb{R}, W \times \mathbb{R})$,

$$\overline{\mathcal{H}}^{\,\sigma(Z \times \mathbb{R})} = \mathcal{N}, \qquad \overline{\mathcal{K}}^{\,\sigma(Y \times \mathbb{R})} = \mathcal{M},$$

so that strong duality and Farkas-type alternatives coincide with weak closedness of the auxiliary sets. In finite dimensions, $\mathcal{H}$ convex and closed is equivalent to its graphical set identity.

7. Illustrative Example: Gale’s Parametrized Program

A canonical infinite-dimensional example (Gale’s parametrized program) considers the primal constraints

$$\sum_{i \geq 0} x_i = \alpha, \quad \sum_{i \geq 1} i x_i = \beta, \quad x_i \geq 0,$$

over sequences $(x_i)$ for prescribed parameters $(\alpha, \beta)$. The dual-side value function satisfies

$$v_{\mathcal{D}}(z_1, z_2) = \begin{cases} 0, & z_1^+ \geq z_2^+ \geq 0 \ +\infty, & \text{otherwise} \end{cases}$$

and hence the epigraphical set

$$\mathcal{N} = \{ (z_1, z_2, t) : z_1^+ \geq z_2^+ \geq 0,\, t \geq 0 \} = \{ (t_1 + t_2^-,\, t_2^-,\, t_3) : t_i \geq 0 \},$$

with $\overline{\mathcal{H}} = \mathcal{N}$ in the weak topology, confirming closedness and strong duality.

8. Concluding Insights and Framework Synthesis

The algebraic framework developed by Pham et al. unifies Farkas-type alternatives and strong duality conditions for infinite-dimensional conic linear problems via the application of perturbed value functions and their graphical/hypergraphical sets, supplanting classical topological requirements (such as Kretschmer’s closedness) by algebraic slice equalities. These slice equalities are both necessary and sufficient for strong duality and for the validity of perturbed Farkas-type lemmas; the classical Farkas lemma is recovered by specializing to the unperturbed case. Algebraic conditions such as the absence of bidual gaps and nonempty cores guarantee the requisite equalities, and topological arguments show that weak closure conditions are equivalent to graphical set identities. This suggests a broad applicability of algebraic separation and perturbation principles throughout infinite-dimensional conic optimization (Khanh et al., 15 Jan 2026).