Invariable plane

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

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 any point $x^{\ast }$ in ${\mathcal {X}}$ for which the number $f(x)$ is smallest, i.e., a point $x^{\ast }$ such that

$f(x^{\ast })\leq f(x)$ for all $x\in {\mathcal {X}}$.

The convexity of $f$ makes the powerful tools of convex analysis applicable. In finite-dimensional normed spaces, the Hahn–Banach theorem and the existence of subgradients lead to a particularly satisfying theory of necessary and sufficient conditions for optimality, a duality theory generalizing that for linear programming, and effective computational methods.

Convex minimization 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 (optimal design), and finance. With recent improvements in computing and in optimization theory, convex minimization is nearly as straightforward as linear programming. Many optimization problems can be reformulated as convex minimization problems. For example, the problem of maximizing a concave function f can be re-formulated equivalently as a problem of minimizing the function -f, which is convex.

Convex optimization problem

An optimization problem (also referred to as a mathematical programming problem or minimization problem) of finding some $x^{\ast }\in {\mathcal {X}}$ such that

$f(x^{\ast })=\min\{f(x):x\in {\mathcal {X}}\},$

where ${\mathcal {X}}\subset \mathbb {R} ^{n}$ is the feasible set and $f(x):\mathbb {R} ^{n}\rightarrow \mathbb {R}$ is the objective, is called convex if ${\mathcal {X}}$ is a closed convex set and $f(x)$ is convex on $\mathbb {R} ^{n}$.

Alternatively, an optimization problem on the form

${\begin{aligned}&\operatorname {minimize} &&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\end{aligned}}$

is called convex if the functions $f,g_{1}\ldots g_{m}:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ are convex.

Theory

The following statements are true about the convex minimization problem:

• if a local minimum exists, then it is a global minimum.
• the set of all (global) minima is convex.
• for each strictly convex function, if the function has a minimum, then the minimum is unique.

These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.

Standard form

Standard form is the usual and most intuitive form of describing a convex minimization 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 terms "linear" and "affine" are often used interchangeably. Such constraints can be expressed in the form $h_{i}(x)=a_{i}^{T}x+b_{i}$, where $a_{i}$ is a column-vector and $b_{i}$ a real number.

A convex minimization problem is thus written as

${\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\\&&&h_{i}(x)=0,\quad i=1,\dots ,p.\end{aligned}}$

Note that every equality constraint $h(x)=0$ can be equivalently replaced by a pair of inequality constraints $h(x)\leq 0$ and $-h(x)\leq 0$. Therefore, for theoretical purposes, equality constraints are redundant; however, it can be beneficial to treat them specially in practice.

Following from this fact, it is easy to understand why $h_{i}(x)=0$ has to be affine as opposed to merely being convex. If $h_{i}(x)$ is convex, $h_{i}(x)\leq 0$ is convex, but $-h_{i}(x)\leq 0$ is concave. Therefore, the only way for $h_{i}(x)=0$ to be convex is for $h_{i}(x)$ to be affine.

Examples

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

• Least squares
• Linear programming
• Convex quadratic minimization with linear constraints
• Quadratically constrained Convex-quadratic minimization with convex quadratic constraints
• Conic optimization
• Geometric programming
• Second order cone programming
• Semidefinite programming
• Entropy maximization with appropriate constraints

Lagrange multipliers

Consider a convex minimization problem given in standard form by a cost function $f(x)$ and inequality constraints $g_{i}(x)\leq 0$, where $i=1\ldots m$. Then the domain ${\mathcal {X}}$ is:

${\mathcal {X}}=\left\lbrace {x\in X\vert g_{1}(x)\leq 0,\ldots ,g_{m}(x)\leq 0}\right\rbrace .$

The Lagrangian 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:

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

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

Convex minimization problems can be solved by the following contemporary methods:

• "Bundle methods" (Wolfe, Lemaréchal, Kiwiel), and
• Subgradient projection methods (Polyak),
• Interior-point methods (Nemirovskii and Nesterov).

Other methods of interest:

• Cutting-plane methods
• Ellipsoid method
• Subgradient method
• Dual subgradients and the drift-plus-penalty method

Subgradient methods can be implemented simply and so are widely used. Dual subgradient methods are subgradient methods applied to a dual problem. The drift-plus-penalty method is similar to the dual subgradient method, but takes a time average of the primal variables.

Convex minimization with good complexity: Self-concordant barriers

The efficiency of iterative methods is poor for the class of convex problems, because this class includes "bad guys" whose minimum cannot be approximated without a large number of function and subgradient evaluations; thus, to have practically appealing efficiency results, it is necessary to make additional restrictions on the class of problems. Two such classes are problems special barrier functions, first self-concordant barrier functions, according to the theory of Nesterov and Nemirovskii, and second self-regular barrier functions according to the theory of Terlaky and coauthors.

Quasiconvex minimization

Problems with convex level sets can be efficiently minimized, in theory. Yuri Nesterov proved that quasi-convex minimization problems could be solved efficiently, and his results were extended by Kiwiel. However, such theoretically "efficient" methods use "divergent-series" stepsize rules, which were first developed for classical subgradient methods. Classical subgradient methods using divergent-series rules are much slower than modern methods of convex minimization, such as subgradient projection methods, bundle methods of descent, and nonsmooth filter methods.

Solving even close-to-convex but non-convex problems can be computationally intractable. Minimizing a unimodal function is intractable, regardless of the smoothness of the function, according to results of Ivanov.

Convex maximization

Conventionally, the definition of the convex optimization problem (we recall) requires that the objective function f to be minimized and the feasible set be convex. In the special case of linear programming (LP), the objective function is both concave and convex, and so LP can also consider the problem of maximizing an objective function without confusion. However, for most convex minimization problems, the objective function is not concave, and therefore a problem and then such problems are formulated in the standard form of convex optimization problems, that is, minimizing the convex objective function.

For nonlinear convex minimization, the associated maximization problem obtained by substituting the supremum operator for the infimum operator is not a problem of convex optimization, as conventionally defined. However, it is studied in the larger field of convex optimization as a problem of convex maximization.

The convex maximization problem is especially important for studying the existence of maxima. Consider the restriction of a convex function to a compact convex set: Then, on that set, the function attains its constrained maximum only on the boundary. Such results, called "maximum principles", are useful in the theory of harmonic functions, potential theory, and partial differential equations.

The problem of minimizing a quadratic multivariate polynomial on a cube is NP-hard. In fact, in the quadratic minimization problem, if the matrix has only one negative eigenvalue, is NP-hard.

Extensions

Advanced treatments consider convex functions that can attain positive infinity, also; the indicator function of convex analysis is zero for every $x\in {\mathcal {X}}$ and positive infinity otherwise.

Extensions of convex functions include biconvex, pseudo-convex, and quasi-convex functions. Partial extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimization problems occur in the field of generalized convexity ("abstract convex analysis").

See also

• Duality
• Karush–Kuhn–Tucker conditions
• Optimization problem
• Proximal Gradient Methods

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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

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

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• Nesterov, Y. and Nemirovsky, A. (1994). 'Interior Point Polynomial Methods in Convex Programming. SIAM

• Nesterov, Yurii. (2004). Introductory Lectures on Convex Optimization, Kluwer Academic Publishers

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Stephen Boyd and Lieven Vandenberghe, Convex optimization (book in pdf)
• EE364a: Convex Optimization I and EE364b: Convex Optimization II, Stanford course homepages
• 6.253: Convex Analysis and Optimization, an MIT OCW course homepage
• Brian Borchers, An overview of software for convex optimization