Convex programming

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Convex optimization is a subfield of mathematical optimization. Given a real vector space $X \$ together with a convex, real-valued function

$f:\mathcal{X}\to \mathbb R$

defined on a convex subset $\mathcal{X}$ of $X \$ , the problem is to find the point $x^* \$ in $\mathcal{X}$ for which the number $f(x) \$ is smallest, i.e., the point $x^* \$ such that $f(x^*) \le f(x)$ for all $x \in X$ .

The convexity of $\mathcal{X}$ and $f \$ make the powerful tools of convex analysis applicable: the Hahn–Banach theorem and the theory of subgradients lead to a particularly satisfying and complete theory of necessary and sufficient conditions for optimality, a duality theory comparable in perfection to that for linear programming, and effective computational methods. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, and finance. Modern computing power has improved the tractability of convex optimization problems to a level roughly equal to that of linear programming.

1 Theory
2 Standard form
3 Examples
4 Methods
5 Software
- 5.1 Convex programming languages
- 5.2 Convex optimization solvers
6 References
7 External links

Theory

The following statements are true about the convex optimization problem:

if a local minimum exists, then it is a global minimum.
the set of all (global) minima is convex.
if the function is strictly convex, then there exists at most one minimum.

The theoretical framework for convex optimization uses the facts above in conjunction with notions from convex analysis such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas's lemma.

Standard form

Standard form is the usual and most intuitive form of describing a convex optimization problem. It consists of the following three parts:

A convex function $f(x): \mathbb{R}^n \to \mathbb{R}$ to be minimized over the variable $x \$
Inequality constraints of the form $g_i(x) \leq 0$ , where the functions $g_i \$ are convex
Equality constraints of the form $h_i(x) = 0 \$ , where the functions $h_i \$ are affine. In practice, the terminology "linear" and "affine" are generally equivalent and most often expressed in the form $h_i(x) = Ax + b \$ , where $A \$ is a matrix and $b \$ is a vector.

A convex optimization problem is thus written as

minimize $\ f(x) \$ subject to

$g_i(x) \leq 0, \quad i = 1,\dots,m$

$h_i(x) = 0, \quad i = 1, \dots,p$

Examples

Hierarchical representation of common convex optimization problems

The following problems are all convex optimization problems, or can be transformed into convex optimization problems via a change of variables:

In a significant example, the set $\mathcal{X}$ is defined by a system of inequalities involving m convex functions g₁, g₂, ..., g_m defined on X, like this:

$\mathcal{X} = \left\lbrace{x\in X \vert g_1(x)\le0, \ldots, g_m(x)\le0}\right\rbrace.$

The Lagrange function for the problem is

L(x,λ₀,...,λ_m) = λ₀f(x) + λ₁g₁(x) + ... + λ_mg_m(x).

For each point x in X that minimizes f over X, there exist real numbers λ₀, ..., λ_m, called Lagrange multipliers, that satisfy these conditions simultaneously:

x minimizes L(y, λ₀, λ₁, ..., λ_m) over all y in X,
λ₀ ≥ 0, λ₁ ≥ 0, ..., λ_m ≥ 0, with at least one λ_k>0,
λ₁g₁(x) = 0, ... , λ_mg_m(x) = 0

If there exists a "strictly feasible point", i.e., a point z satisfying

g₁(z) < 0,...,g_m(z) < 0,

then the statement above can be upgraded to assert that λ₀=1.

Conversely, if some x in X satisfies 1)-3) for scalars λ₀, ..., λ_m with λ₀ = 1, then x is certain to minimize f over X.

Methods

The following methods are used in solving convex optimization problems:

Software

Although most general-purpose nonlinear equation solvers such as LSSOL, LOQO, MINOS, and Lancelot work well, many software packages dealing exclusively with convex optimization problems are also available:

Convex programming languages

Convex optimization solvers

References

Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press.

Luenberger, David (1984). Linear and Nonlinear Programming. Addison-Wesley.

Luenberger, David (1969). Optimization by Vector Space Methods. Wiley & Sons.

Bertsekas, Dimitri (2003). Convex Analysis and Optimization. Athena Scientific.

Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.

Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.

Berkovitz, Leonard (2001). Convexity and Optimization in $\mathbb{R}^n$ . John Wiley & Sons.

Ruszczynski, Andrzej (2006). Nonlinear Optimization. Princeton University Press.

S.I. Zukhovitsky, L.I. Avdeeva, "Linear and convex programming" , Saunders (1966)

External links

Stephen Boyd and Lieven Vandenberghe, Convex Optimization (book in pdf)
EE364 and EE364b, a Stanford course homepage
6.253, an MIT OCW course homepage
Jon Dattorro, Convex Optimization & Euclidean Distance Geometry pdf
Haitham Hindi, A Tutorial on Convex Optimization This has an slight engineering focus but is written informally and at a very accessible level.