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.
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, finance, statistics (optimal experimental design), and structural optimization, where the approximation concept has proven to be efficient. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming.
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 mapping some subset of into is convex if its domain is convex and for all and all in its domain, the following condition holds: . A set S is convex if for all members and all , we have that .
Concretely, a convex optimization problem is the problem of finding some attaining
where the objective function is convex, as is the feasible set .  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 is unbounded below over or the infimum is not attained, then the optimization problem is said to be unbounded. Otherwise, if is the empty set, then the problem is said to be infeasible.
A convex optimization problem is in standard form if it is written as
This notation describes the problem of finding that minimizes among all satisfying , and , . The function is the objective function of the problem, and the functions and are the constraint functions.
The feasible set of the optimization problem consists of all points satisfying the constraints. This set is convex because is convex, the sublevel sets of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex.
A solution to a convex optimization problem is any point attaining . In general, a convex optimization problem may have zero, one, or many solutions.
Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function can be re-formulated equivalently as the problem of minimizing the convex function . The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.
The following are useful properties of convex optimization problems:
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.
The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations:
Convex optimization has practical applications for the following.
Consider a convex minimization problem given in standard form by a cost function and inequality constraints for . Then the domain is:
The Lagrangian function for the problem is
For each point in that minimizes over , there exist real numbers called Lagrange multipliers, that satisfy these conditions simultaneously:
If there exists a "strictly feasible point", that is, a point satisfying
then the statement above can be strengthened to require that .
Conversely, if some in satisfies (1)–(3) for scalars with then is certain to minimize over .
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. 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. 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. (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.)
Convex optimization problems can also be solved by the following contemporary methods:
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.
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.
|CVX||MATLAB||Interfaces with SeDuMi and SDPT3 solvers; designed to only express convex optimization problems.||Yes|||
|CVXMOD||Python||Interfaces with the CVXOPT solver.||Yes|||
|Convex.jl||Julia||Disciplined convex programming, supports many solvers.||Yes|||
|YALMIP||MATLAB, Octave||Interfaces with CPLEX, GUROBI, MOSEK, SDPT3, SEDUMI, CSDP, SDPA, PENNON solvers; also supports integer and nonlinear optimization, and some nonconvex optimization. Can perform robust optimization with uncertainty in LP/SOCP/SDP constraints.||Yes|||
|LMI lab||MATLAB||Expresses and solves semidefinite programming problems (called "linear matrix inequalities")||No|||
|LMIlab translator||Transforms LMI lab problems into SDP problems.||Yes|||
|xLMI||MATLAB||Similar to LMI lab, but uses the SeDuMi solver.||Yes|||
|AIMMS||Can do robust optimization on linear programming (with MOSEK to solve second-order cone programming) and mixed integer linear programming. Modeling package for LP + SDP and robust versions.||No|||
|ROME||Modeling system for robust optimization. Supports distributionally robust optimization and uncertainty sets.||Yes|||
|Modeling system for polynomial optimization.||Yes|||
|SOSTOOLS||Modeling system for polynomial optimization. Uses SDPT3 and SeDuMi. Requires Symbolic Computation Toolbox.||Yes|||
|SparsePOP||Modeling system for polynomial optimization. Uses the SDPA or SeDuMi solvers.||Yes|||
|CPLEX||Supports primal-dual methods for LP + SOCP. Can solve LP, QP, SOCP, and mixed integer linear programming problems.||No|||
|CSDP||C||Supports primal-dual methods for LP + SDP. Interfaces available for MATLAB, R, and Python. Parallel version available. SDP solver.||Yes|||
|CVXOPT||Python||Supports primal-dual methods for LP + SOCP + SDP. Uses Nesterov-Todd scaling. Interfaces to MOSEK and DSDP.||Yes|||
|MOSEK||Supports primal-dual methods for LP + SOCP.||No|||
|SeDuMi||MATLAB, Octave, MEX||Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP.||Yes|||
|SDPA||C++||Solves LP + SDP. Supports primal-dual methods for LP + SDP. Parallelized and extended precision versions are available.||Yes|||
|SDPT3||MATLAB, Octave, MEX||Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP.||Yes|||
|ConicBundle||Supports general-purpose codes for LP + SOCP + SDP. Uses a bundle method. Special support for SDP and SOCP constraints.||Yes|||
|DSDP||Supports general-purpose codes for LP + SDP. Uses a dual interior point method.||Yes|||
|LOQO||Supports general-purpose codes for SOCP, which it treats as a nonlinear programming problem.||No|||
|PENNON||Supports general-purpose codes. Uses an augmented Lagrangian method, especially for problems with SDP constraints.||No|||
|SDPLR||Supports general-purpose codes. Uses low-rank factorization with an augmented Lagrangian method.||Yes|||
|GAMS||Modeling system for linear, nonlinear, mixed integer linear/nonlinear, and second-order cone programming problems.||No|||
|Optimization Services||XML standard for encoding optimization problems and solutions.|||
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.
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