Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

== Definition ==

=== Abstract form === A convex optimization problem is defined by two ingredients:

The objective function, which is a real-valued convex function of n variables, $f:{\mathcal {D}}\subseteq \mathbb {R} ^{n}\to \mathbb {R}$
; The feasible set, which is a convex subset $C\subseteq \mathbb {R} ^{n}$
. The goal of the problem is to find some $\inf\{f(\mathbf {x} ):\mathbf {x} \in C\}$
. In general, there are three options regarding the existence of a solution:

If such a point x* exists, it is referred to as an optimal point or solution; the set of all optimal points is called the optimal set; and the problem is called solvable. If $C$
, or the infimum is not attained, then the optimization problem is said to be unbounded. Otherwise, if $C$

=== Standard form === A convex optimization problem is in standard form if it is written as $\mathbf {x} \in \mathbb {R} ^{n}$ The objective function $f:{\mathcal {D}}\subseteq \mathbb {R} ^{n}\to \mathbb {R}$ The inequality constraint functions $g_{i}:\mathbb {R} ^{n}\to \mathbb {R}$
, $i=1,\ldots ,m$
, are convex functions; The equality constraint functions $h_{i}:\mathbb {R} ^{n}\to \mathbb {R}$
, $i=1,\ldots ,p$
, are affine transformations, that is, of the form: $h_{i}(\mathbf {x} )=\mathbf {a_{i}} \cdot \mathbf {x} -b_{i}$
, where $b_{i}$ The feasible set ${\mathcal {D}}$ Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function $-f$
. The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.

=== Epigraph form (standard form with linear objective) === In the standard form it is possible to assume, without loss of generality, that the objective function f is a linear function. This is because any program with a general objective can be transformed into a program with a linear objective by adding a single variable t and a single constraint, as follows: ${\begin{aligned}&{\underset {\mathbf {x} ,t}{\operatorname {minimize} }}&&t\\&\operatorname {subject\ to} &&f(\mathbf {x} )-t\leq 0\\&&&g_{i}(\mathbf {x} )\leq 0,\quad i=1,\dots ,m\\&&&h_{i}(\mathbf {x} )=0,\quad i=1,\dots ,p,\end{aligned}}$ Every convex program can be presented in a conic form, which means minimizing a linear objective over the intersection of an affine plane and a convex cone: ${\begin{aligned}&{\underset {\mathbf {x} }{\operatorname {minimize} }}&&c^{T}x\\&\operatorname {subject\ to} &&x\in (b+L)\cap K\end{aligned}}$

=== Eliminating linear equality constraints === It is possible to convert a convex program in standard form, to a convex program with no equality constraints. Denote the equality constraints hi(x)=0 as Ax=b, where A has n columns. If Ax=b is infeasible, then of course the original problem is infeasible. Otherwise, it has some solution x0 , and the set of all solutions can be presented as: Fz+x0, where z is in Rk, k=n-rank(A), and F is an n-by-k matrix. Substituting x = Fz+x0 in the original problem gives: ${\begin{aligned}&{\underset {\mathbf {x} }{\operatorname {minimize} }}&&f(\mathbf {F\mathbf {z} +\mathbf {x} _{0}} )\\&\operatorname {subject\ to} &&g_{i}(\mathbf {F\mathbf {z} +\mathbf {x} _{0}} )\leq 0,\quad i=1,\dots ,m\\\end{aligned}}$
where the variables are z. Note that there are rank(A) fewer variables. This means that, in principle, one can restrict attention to convex optimization problems without equality constraints. In practice, however, it is often preferred to retain the equality constraints, since they might make some algorithms more efficient, and also make the problem easier to understand and analyze.

== Special cases == The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations:

Linear programming problems are the simplest convex programs. In LP, the objective and constraint functions are all linear. Quadratic programming are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function. Second order cone programming are more general. Semidefinite programming are more general. Conic optimization are even more general - see figure to the right, Other special cases include;

Least squares Quadratic minimization with convex quadratic constraints Geometric programming Entropy maximization with appropriate constraints.

== Properties == The following are useful properties of convex optimization problems:

every point that is local minimum is also a global minimum; the optimal set is convex; if the objective function is strictly convex, then the problem has at most one optimal point. 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.

== Algorithms ==

=== Unconstrained and equality-constrained problems === The convex programs easiest to solve are the unconstrained problems, or the problems with only equality constraints. As the equality constraints are all linear, they can be eliminated with linear algebra and integrated into the objective, thus converting an equality-constrained problem into an unconstrained one. In the class of unconstrained (or equality-constrained) problems, the simplest ones are those in which the objective is quadratic. For these problems, the KKT conditions (which are necessary for optimality) are all linear, so they can be solved analytically. For unconstrained (or equality-constrained) problems with a general convex objective that is twice-differentiable, Newton's method can be used. It can be seen as reducing a general unconstrained convex problem, to a sequence of quadratic problems.Newton's method can be combined with line search for an appropriate step size, and it can be mathematically proven to converge quickly. Other efficient algorithms for unconstrained minimization are gradient descent (a special case of steepest descent).

=== General problems === The more challenging problems are those with inequality constraints. A common way to solve them is to reduce them to unconstrained problems by adding a barrier function, enforcing the inequality constraints, to the objective function. Such methods are called interior point methods.They have to be initialized by finding a feasible interior point using by so-called phase I methods, which either find a feasible point or show that none exist. Phase I methods generally consist of reducing the search in question to a simpler convex optimization problem. Convex optimization problems can also be solved by the following contemporary methods:

Bundle methods (Wolfe, Lemaréchal, Kiwiel), and Subgradient projection methods (Polyak), Interior-point methods, which make use of self-concordant barrier functions and self-regular barrier functions. 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.

== Lagrange multipliers ==

Consider a convex minimization problem given in standard form by a cost function $1\leq i\leq m$
. Then the domain ${\mathcal {X}}=\left\{x\in X\vert g_{1}(x),\ldots ,g_{m}(x)\leq 0\right\}.$ (Article truncated for display)

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Category

Optimization - Mathematics