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,^[1] whereas mathematical optimization is in general NP-hard.^[2][3][4]

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,^[5] data analysis and modeling, finance, statistics (optimal experimental design),^[6] and structural optimization, where the approximation concept has proven to be efficient.^[7][8] With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming.^[9]

Definition

A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set is a convex set. A function ${\displaystyle f}$ mapping some subset of ${\displaystyle \mathbb {R} ^{n))$into ${\displaystyle \mathbb {R} \cup \{\pm \infty \))$ is convex if its domain is convex and for all ${\displaystyle \theta \in [0,1]}$ and all ${\displaystyle x,y}$ in its domain, the following condition holds: ${\displaystyle f(\theta x+(1-\theta )y)\leq \theta f(x)+(1-\theta )f(y)}$. A set S is convex if for all members ${\displaystyle x,y\in S}$ and all ${\displaystyle \theta \in [0,1]}$, we have that ${\displaystyle \theta x+(1-\theta )y\in S}$.

Concretely, a convex optimization problem is the problem of finding some ${\displaystyle \mathbf {x^{\ast )) \in C}$ attaining

${\displaystyle \inf\{f(\mathbf {x} ):\mathbf {x} \in C\))$,

where the objective function ${\displaystyle f:{\mathcal {D))\subseteq \mathbb {R} ^{n}\to \mathbb {R} }$ is convex, as is the feasible set ${\displaystyle C}$.^{[10] [11]} If such a point exists, it is referred to as an optimal point or solution; the set of all optimal points is called the optimal set. If ${\displaystyle f}$ is unbounded below over ${\displaystyle C}$ or the infimum is not attained, then the optimization problem is said to be unbounded. Otherwise, if ${\displaystyle C}$ is the empty set, then the problem is said to be infeasible.^[12]

Standard form

A convex optimization problem is in standard form if it is written as

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

where:^[12]

• ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n))$ is the optimization variable;
• The objective function ${\displaystyle f:{\mathcal {D))\subseteq \mathbb {R} ^{n}\to \mathbb {R} }$ is a convex function;
• The inequality constraint functions ${\displaystyle g_{i}:\mathbb {R} ^{n}\to \mathbb {R} }$, ${\displaystyle i=1,\ldots ,m}$, are convex functions;
• The equality constraint functions ${\displaystyle h_{i}:\mathbb {R} ^{n}\to \mathbb {R} }$, ${\displaystyle i=1,\ldots ,p}$, are affine transformations, that is, of the form: ${\displaystyle h_{i}(\mathbf {x} )=\mathbf {a_{i)) \cdot \mathbf {x} -b_{i))$, where ${\displaystyle \mathbf {a_{i)) }$ is a vector and ${\displaystyle b_{i))$ is a scalar.

This notation describes the problem of finding ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n))$ that minimizes ${\displaystyle f(\mathbf {x} )}$ among all ${\displaystyle \mathbf {x} }$ satisfying ${\displaystyle g_{i}(\mathbf {x} )\leq 0}$, ${\displaystyle i=1,\ldots ,m}$ and ${\displaystyle h_{i}(\mathbf {x} )=0}$, ${\displaystyle i=1,\ldots ,p}$. The function ${\displaystyle f}$ is the objective function of the problem, and the functions ${\displaystyle g_{i))$ and ${\displaystyle h_{i))$ are the constraint functions.

The feasible set ${\displaystyle C}$ of the optimization problem consists of all points ${\displaystyle \mathbf {x} \in {\mathcal {D))}$ satisfying the constraints. This set is convex because ${\displaystyle {\mathcal {D))}$ is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex.^[13]

A solution to a convex optimization problem is any point ${\displaystyle \mathbf {x} \in C}$ attaining ${\displaystyle \inf\{f(\mathbf {x} ):\mathbf {x} \in C\))$. In general, a convex optimization problem may have zero, one, or many solutions.^[14]

Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function ${\displaystyle f}$ can be re-formulated equivalently as the problem of minimizing the convex function ${\displaystyle -f}$. The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.^[15]

Properties

The following are useful properties of convex optimization problems:^[16][12]

• every local minimum is 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.^{[citation needed]}

Applications

The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations:^[12][17]

A hierarchy of convex optimization problems. (LP: linear program, QP: quadratic program, SOCP second-order cone program, SDP: semidefinite program, CP: cone program.)

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

Convex optimization has practical applications for the following.

• Portfolio optimization.^[18]
• Worst-case risk analysis.^[18]
• Optimal advertising.^[18]
• Variations of statistical regression (including regularization and quantile regression).^[18]
• Model fitting^[18] (particularly multiclass classification^[19]).
• Electricity generation optimization.^[19]
• Combinatorial optimization.^[19]
• Non-probabilistic modelling of uncertainty.^[20]
• Localization using wireless signals ^[21]

Lagrange multipliers

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

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

The Lagrangian function for the problem is

${\displaystyle L(x,\lambda _{0},\lambda _{1},\ldots ,\lambda _{m})=\lambda _{0}f(x)+\lambda _{1}g_{1}(x)+\cdots +\lambda _{m}g_{m}(x).}$

For each point ${\displaystyle x}$ in ${\displaystyle X}$ that minimizes ${\displaystyle f}$ over ${\displaystyle X}$, there exist real numbers ${\displaystyle \lambda _{0},\lambda _{1},\ldots ,\lambda _{m},}$ called Lagrange multipliers, that satisfy these conditions simultaneously:

1. ${\displaystyle x}$ minimizes ${\displaystyle L(y,\lambda _{0},\lambda _{1},\ldots ,\lambda _{m})}$ over all ${\displaystyle y\in X,}$
2. ${\displaystyle \lambda _{0},\lambda _{1},\ldots ,\lambda _{m}\geq 0,}$ with at least one ${\displaystyle \lambda _{k}>0,}$
3. ${\displaystyle \lambda _{1}g_{1}(x)=\cdots =\lambda _{m}g_{m}(x)=0}$ (complementary slackness).

If there exists a "strictly feasible point", that is, a point ${\displaystyle z}$ satisfying

${\displaystyle g_{1}(z),\ldots ,g_{m}(z)<0,}$

then the statement above can be strengthened to require that ${\displaystyle \lambda _{0}=1}$.

Conversely, if some ${\displaystyle x}$ in ${\displaystyle X}$ satisfies (1)–(3) for scalars ${\displaystyle \lambda _{0},\ldots ,\lambda _{m))$ with ${\displaystyle \lambda _{0}=1}$ then ${\displaystyle x}$ is certain to minimize ${\displaystyle f}$ over ${\displaystyle X}$.

Algorithms

Unconstrained convex optimization can be easily solved with gradient descent (a special case of steepest descent) or Newton's method, combined with line search for an appropriate step size; these can be mathematically proven to converge quickly, especially the latter method.^[22] Convex optimization with linear equality constraints can also be solved using KKT matrix techniques if the objective function is a quadratic function (which generalizes to a variation of Newton's method, which works even if the point of initialization does not satisfy the constraints), but can also generally be solved by eliminating the equality constraints with linear algebra or solving the dual problem.^[22] Finally, convex optimization with both linear equality constraints and convex inequality constraints can be solved by applying an unconstrained convex optimization technique to the objective function plus logarithmic barrier terms.^[22] (When the starting point is not feasible - that is, satisfying the constraints - this is preceded 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 yet another convex optimization problem.^[22])

Convex optimization problems can also be solved by the following contemporary methods:^[23]

• Bundle methods (Wolfe, Lemaréchal, Kiwiel), and
• Subgradient projection methods (Polyak),
• Interior-point methods,^[1] which make use of self-concordant barrier functions ^[24] and self-regular barrier functions.^[25]
• 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.^{[26][citation needed]} 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.^{[citation needed]}

Software

There is a large software ecosystem for convex optimization. This ecosystem has two main categories: solvers on the one hand and modeling tools (or interfaces) on the other hand.

Solvers implement the algorithms themselves and are usually written in C. They require users to specify optimization problems in very specific formats which may not be natural from a modeling perspective. Modeling tools are separate pieces of software that let the user specify an optimization in higher-level syntax. They manage all transformations to and from the user's high-level model and the solver's input/output format.

The table below shows a mix of modeling tools (such as CVXPY and Convex.jl) and solvers (such as CVXOPT and MOSEK). This table is by no means exhaustive.

Extensions

Extensions of convex optimization include the optimization of biconvex, pseudo-convex, and quasiconvex functions. Extensions of the theory of convex analysis and iterative methods for approximately solving non-convex minimization problems occur in the field of generalized convexity, also known as abstract convex analysis.^{[citation needed]}