What does "curly (curved) less than" sign $\succcurlyeq$ mean?

Question

I am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)

$$f(x^*) \succcurlyeq \alpha$$

and curved less than or equal to (\preccurlyeq)

$$f(x^*) \preccurlyeq \alpha$$

Can someone explain what they mean?

Answer 1

Both Chris Culter's and Code Guru's answers are good, and I've voted them both up. I hope that I'm not being inappropriate by combining and expanding upon them here.

It should be noted that the book does not use $\succeq$, $\preceq$, $\succ$, and $\prec$ with scalar inequalities; for these, good old-fashioned inequality symbols suffice. It is only when the quantities on the left- and right-hand sides are vectors, matrices, or other multi-dimensional objects that this notation is called for.

The book refers to these relations as generalized inequalities, but as Code-Guru rightly points out, they have been in use for some time to represent partial orderings. And indeed, that's exactly what they are, and the book does refer to them that way as well. But given that the text deals with convex optimization, it was apparently considered helpful to refer to them as inequalities.

Let $S$ be a vector space, and let $K\subset S$ be a closed, convex, and pointed cone with a non-empty interior. (By cone, we mean that $\alpha K\equiv K$ for all $\alpha>0$; and by pointed, we mean that $K\cap-K=\{0\}$.) Such a cone $K$ induces a partial ordering on the set $S$, and an associated set of generalized inequalities: $$ x \succeq_K y \quad\Longleftrightarrow\quad y \preceq_K x \quad\Longleftrightarrow\quad x - y \in K $$ $$ x \succ_K y \quad\Longleftrightarrow\quad y \prec_K x \quad\Longleftrightarrow\quad x - y \in \mathop{\textrm{Int}} K $$ This is a partial ordering because, for many pairs $x,y\in S$, $x \not\succeq_K y$ and $y \not\succeq_K x$. So that's the primary reason why he and others prefer to use the curly inequalities to denote these orderings, reserving $\geq$, $\leq$, etc. for total orderings. But it has many of the properties of a standard inequality, such as: $$x\succeq_K y \quad\Longrightarrow\quad \alpha x \succeq_K \alpha y \quad\forall \alpha>0$$ $$x\succeq_K y \quad\Longrightarrow\quad \alpha x \preceq_K \alpha y \quad\forall \alpha<0$$ $$x\succeq_K y, ~ x\preceq_K y \quad\Longrightarrow\quad x=y$$ $$x\succ_K y \quad\Longrightarrow\quad x\not\prec_K y$$

When the cone $K$ is understood from context, it is often dropped, leaving only the inequality symbol $\succeq$. There are two cases where this is almost always done. First, when $S=\mathbb{R}^n$ and the cone $K$ is non-negative orthant $\mathbb{R}^n_+$ the generalized inequality is simply an elementwise inequality: $$x \succeq_{\mathbb{R}^n_+} y \quad\Longleftrightarrow\quad x_i\geq y_i,~i=1,2,\dots,n$$ Second, when $S$ is the set of symmetric $n\times n$ matrices and $K$ is the cone of positive semidefinite matrices $\mathcal{S}^n_+=\{X\in S\,|\,\lambda_{\text{min}}(X)\geq 0\}$, the inequality is a linear matrix inequality (LMI): $$X \succeq_{\mathcal{S}^n_+} Y \quad\Longleftrightarrow\quad \lambda_{\text{min}}(X-Y)\geq 0$$ In both of these cases, the cone subscript is almost always dropped.

Many texts in convex optimization don't bother with this distinction, and use $\geq$ and $\leq$ even for LMIs and other partial orderings. I prefer to use it whenever I can, because I think it helps people realize that this is not a standard inequality with an underlying total order. That said, I don't feel that strongly about it for $\mathbb{R}^n_+$; I think most people rightly assume that $x\geq y$ is considered elementwise when $x,y$ are vectors.

Answer 2

There's a list of notation in the back of the book. On page 698, $x\preceq y$ is defined as componentwise inequality between vectors $x$ and $y$. This means that $x_i\leq y_i$ for every index $i$.

Edit: The notation is introduced on page 32.

Answer 3

Often these symbols represent partial order relations. The typical "less than" and "greater than" operations both define partial orders on the real numbers. However, there are many other examples of partial orders.

Answer 4

Sometimes the curly greater than sign is used to indicate positive semi-definiteness of a matrix $X$:

$$(X\succeq 0\ \text{or}\ X\ge 0)$$

or a function $f(x)$

$$ (f(x) \succeq 0\ \text{or}\ f(x)\ge 0). $$

Answer 5

Does it sometimes denote, in measurement theory; often the qualitative counterpart to $\geq $ in the numerical representation; when one wants to numerically represent a totally ordered qualitative probability representation:

$$ A ≽ B \leftrightarrow A > B \leftrightarrow F(A) \geq F(B)$$.

On the other hand. I have seen it used in multi-dimensional partial or even total, orderings under the "ordering of major-ization" for vector valued functions or for a system for two kinds of orderings.

One for (numerical) ordinal, $>$ comparisons and another, $≽$ for (numerical) differences, sums, or a some other kind of relation, to fine grain, the representation to ensure (or some kind of) unique-ness, rather than merely strong represent-ability.

See Marshall

Marshall, Albert W.; Olkin, Ingram, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, Vol. 143. New York etc.: Academic Press. XX, 569 p. \$ 52.50 (1979). ZBL0437.26007.

Such functions or representations, may use both the curly greater $≽$ than and $\geq $ in the numerical or functional representation and are compatible with total orderings.

$$a,b\in \Omega^{n}\; a <b \,,\quad \text{or}, \quad, a= b, \quad \text{or}\quad a < b$$

where one cannot usually adding up distinct $$a_1 \in \Omega_i\,; b_j\in \Omega_j; j \neq i$$

$$a_3 \in \Omega_c;\, ; \, P(a_i) + P(b_j) = P(a_3)$$.

Sometimes I "(conjecture)" that $≽$ denotes a weaker version of $(A)$ or $(A_1)$ (strong complementary add-itivity) which orders the differences and sums, is something over and above

$(B)$;strict monotone increasing/strong representability/order embedding- above reflecting and preserving .

$$(A)\,[f(x_1)+f(x_2)≽ f(y_1) + f(y_2)] \rightarrow [f(x) \geq f(y)] \text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$ or $$(A1)\,[x\, \geq y ]\,\rightarrow [f(x_1)+f(x_2)≽ f(y_1) + f(y_2)]\text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$

Rather than:$$(B)$$

$$(B)\,x\, \geq y \leftrightarrow f(x) \geq f(y)\, \text{where} \,; x=x_1 +x_2\,, y=y_1+y_2 $$

In contrast to $(B)$ a standard order embedding (strict monotone increasing function)

Where in $(B)$ the numerical function, $F$ or representation is 'merely strong'. That is $F$ is a monotone strictly increasing function of some entity $x$ where $F(x)$ is the entity one wishes to order by $x$ .

Sometimes, even in a infinite and uniformly and non-atomic, continuous total order representation, nothing unique will come out. If its infinite /non-atomic in the wrong sense.

ie in a super-atomic or entangled lexicographic system. Where the system is dense/or continuum dense/non-atomic/bottom-less in the wrong sense. Such as a spin $1/2$ system in quantum mechanics, or a lexicographic entangled, system where its infinite in wrong sense, in the vertical, not in the horizontal, not within a basis of space/basis.

$(A)$ is arguably a lot stronger than $(B)$, or at least is, if relatively unrestricted.

Where $(A)$, may be used not to extend the ordering so much as much as make a total ordering 'numerically precise; to put some constraints on sums and differences, where the events are the not kinds of things that usually add up.

Say in an entangled multidimensional quantum spin 1/2 system or utility representation where the events are on complementary spaces and mixtures are not allowed, for example.

That is, to, put a metric on differences (say on a two outcome system). That is something stronger, than a, total or non atomic/dense order, where(merely) every event.

Even if the entire system is totally ordered within and betwixt the distinct $\Omega$. So that the system is solv-able, and uniquely so.