Conjugate set

1 Conjugate (Fenchel conjugate, dual, Fenchel dual) set

1.1 Definitions

Let S \subseteq \mathbb{R}^n be an arbitrary non-empty set. Then its conjugate set is defined as:

S^* = \{y \in \mathbb{R}^n \mid \langle y, x\rangle \ge -1 \;\; \forall x \in S\}

Figure 1: Convex sets may be described in a dual way - through the elements of the set and through the set of hyperplanes supporting it

A set S^{**} is called double conjugate to a set S if:

S^{**} = \{y \in \mathbb{R}^n \mid \langle y, x\rangle \ge -1 \;\; \forall x \in S^*\}

  • The sets S_1 and S_2 are called inter-conjugate if S_1^* = S_2, S_2^* = S_1.
  • A set S is called self-conjugate if S^{*} = S.

1.2 Properties

  • A conjugate set is always closed, convex, and contains zero.

  • For an arbitrary set S \subseteq \mathbb{R}^n:

    S^{**} = \overline{ \mathbf{conv} (S \cup \{0\}) }

  • If S_1 \subseteq S_2, then S_2^* \subseteq S_1^*.

  • \left( \bigcup\limits_{i=1}^m S_i \right)^* = \bigcap\limits_{i=1}^m S_i^*.

  • If S is closed, convex, and includes 0, then S^{**} = S.

  • S^* = \left(\overline{S}\right)^*.

1.3 Examples

Example

Prove that S^* = \left(\overline{S}\right)^*.

  • S \subset \overline{S}\rightarrow \left(\overline{S}\right)^* \subset S^*.
  • Let p \in S^* and x_0 \in \overline{S}, x_0 = \underset{k \to \infty}{\operatorname{lim}} x_k. Then by virtue of the continuity of the function f(x) = p^Tx, we have: p^T x_k \ge -1 \to p^Tx_0 \ge -1. So p \in \left(\overline{S}\right)^*, hence S^* \subset \left(\overline{S}\right)^*.
Example

Prove that \left( \mathbf{conv}(S) \right)^* = S^*.

  • S \subset \mathbf{conv}(S) \to \left( \mathbf{conv}(S) \right)^* \subset S^*.

  • Let p \in S^*, x_0 \in \mathbf{conv}(S), i.e., x_0 = \sum\limits_{i=1}^k\theta_i x_i \mid x_i \in S, \sum\limits_{i=1}^k\theta_i = 1, \theta_i \ge 0.

    So p^T x_0 = \sum\limits_{i=1}^k\theta_i p^Tx_i \ge \sum\limits_{i=1}^k\theta_i (-1) = 1 \cdot (-1) = -1. So p \in \left( \mathbf{conv}(S) \right)^*, hence S^* \subset \left( \mathbf{conv}(S) \right)^*.

Example

Prove that if B(0,r) is a ball of radius r by some norm centered at zero, then \left( B(0,r) \right)^* = B(0,1/r).

  • Let B(0,r) = X, B(0,1/r) = Y. Take the normal vector p \in X^*, then for any x \in X: p^Tx \ge -1.

  • From all points of the ball X, take such a point x \in X that its scalar product of p: p^Tx is minimal, then this is the point x = -\frac{p}{\|p\|}r.

    p^T x = p^T \left(-\frac{p}{\|p\||}r \right) = -\|p\|r \ge -1

    \|p\| \le \frac{1}{r} \in Y

    So X^* \subset Y.

  • Now let p \in Y. We need to show that p \in X^*, i.e., \langle p, x\rangle \geq -1. It’s enough to apply the Cauchy-Bunyakovsky inequality:

    \|\langle p, x\rangle\| \leq \|p\| \|x\| \leq \dfrac{1}{r} \cdot r = 1

    The latter comes from the fact that p \in B(0,1/r) and x \in B(0,r).

    So Y \subset X^*.

1.4 Dual cones

A conjugate cone to a cone K is a set K^* such that:

K^* = \left\{ y \mid \langle x, y\rangle \ge 0 \quad \forall x \in K\right\}

To show that this definition follows directly from the theorem above, recall what a conjugate set is and what a cone \forall \lambda > 0 is.

\{y \in \mathbb{R}^n \mid \langle y, x\rangle \ge -1 \;\; \forall x \in S\} \to \to \{\lambda y \in \mathbb{R}^n \mid \langle y, x\rangle \ge -\dfrac{1}{\lambda} \;\; \forall x\in S\}

Figure 2: Illustration of dual cone

1.5 Dual cones properties

  • Let K be a closed convex cone. Then K^{**} = K.

  • For an arbitrary set S \subseteq \mathbb{R}^n and a cone K \subseteq \mathbb{R}^n:

    \left( S + K \right)^* = S^* \cap K^*

  • Let K_1, \ldots, K_m be cones in \mathbb{R}^n, then:

    \left( \sum\limits_{i=1}^m K_i \right)^* = \bigcap\limits_{i=1}^m K_i^*

  • Let K_1, \ldots, K_m be cones in \mathbb{R}^n. Let also their intersection have an interior point, then:

    \left( \bigcap\limits_{i=1}^m K_i \right)^* = \sum\limits_{i=1}^m K_i^*

1.6 Examples

Example

Find the conjugate cone for a monotone nonnegative cone:

K = \left\{ x \in \mathbb{R}^n \mid x_1 \ge x_2 \ge \ldots \ge x_n \ge 0\right\}

Note that:

\sum\limits_{i=1}^nx_iy_i = y_1 (x_1-x_2) + (y_1 + y_2)(x_2 - x_3) + \ldots + (y_1 + y_2 + \ldots + y_{n-1})(x_{n-1} - x_n) + (y_1 + \ldots + y_n)x_n

Since in the presented sum in each summand, the second multiplier in each summand is non-negative, then:

y_1 \ge 0, \;\; y_1 + y_2 \ge 0, \;\;\ldots, \;\;\;y_1 + \ldots + y_n \ge 0

So K^* = \left\{ y \mid \sum\limits_{i=1}^k y_i \ge 0, k = \overline{1,n}\right\}.

1.7 Polyhedra

The set of solutions to a system of linear inequalities and equalities is a polyhedron:

Ax \preceq b, \;\;\; Cx = d

Here A \in \mathbb{R}^{m\times n}, C \in \mathbb{R}^{p \times n}, and the inequality is a piecewise inequality.

Figure 3: Polyhedra
Theorem

Let x_1, \ldots, x_m \in \mathbb{R}^n. Conjugate to a polyhedral set:

S = \mathbf{conv}(x_1, \ldots, x_k) + \mathbf{cone}(x_{k+1}, \ldots, x_m)

is a polyhedron (polyhedron):

S^* = \left\{ p \in \mathbb{R}^n \mid \langle p, x_i\rangle \ge -1, i = \overline{1,k} ; \langle p, x_i\rangle \ge 0, i = \overline{k+1,m} \right\}

  • Let S = X, S^* = Y. Take some p \in X^*, then \langle p, x_i\rangle \ge -1, i = \overline{1,k}. At the same time, for any \theta > 0, i = \overline{k+1,m}:

    \langle p, x_i\rangle \ge -1 \to \langle p, \theta x_i\rangle \ge -1

    \langle p, x_i\rangle \ge -\frac{1}{\theta} \to \langle p, x_i\rangle \geq 0.

    So p \in Y \to X^* \subset Y.

  • Suppose, on the other hand, that p \in Y. For any point x \in X:

    x = \sum\limits_{i=1}^m\theta_i x_i \;\;\;\;\;\;\; \sum\limits_{i=1}^k\theta_i = 1, \theta_i \ge 0

    So:

    \langle p, x\rangle = \sum\limits_{i=1}^m\theta_i \langle p, x_i\rangle = \sum\limits_{i=1}^k\theta_i \langle p, x_i\rangle + \sum\limits_{i=k+1}^m\theta_i \langle p, x_i\rangle \ge \sum\limits_{i=1}^k\theta_i (-1) + \sum\limits_{i=1}^k\theta_i \cdot 0 = -1.

    So p \in X^* \to Y \subset X^*.

Example

Find and represent the set conjugate to a polyhedral cone in the plane:

S = \mathbf{cone} \left\{ (-3,1), (2,3), (4,5)\right\}

Using the theorem above:

S^* = \left\{ -3p_1 + p_2 \ge 0, 2p_1 + 3p_2 \ge 0, 4p_1+5p_2 \ge 0 \right\}

1.7.1 Farkas’ Lemma

Let A \in \mathbb{R}^{m\times n}, b \in \mathbb{R}^m. Then one and only one of the following two systems has a solution:

1) \; Ax = b, x \ge 0

2) \; p^\top A \ge 0, \langle p,b\rangle < 0.

Ax = b when x \geq 0 means that b lies in a cone stretched over the columns of the matrix A.

pA \geq 0, \; \langle p, b \rangle < 0 means that there exists a separating hyperplane between the vector b and the cone of columns of the matrix A.

1.7.1.1 Corollary:

Let A \in \mathbb{R}^{m\times n}, b \in \mathbb{R}^m. Then one and only one of the following two systems has a solution:

1) Ax \leq b

2) p^\top A = 0, \langle p,b\rangle < 0, p \ge 0.

If in the minimization linear programming problem, the budget set is non-empty and the target function is bounded on it from below, then the problem has a solution.