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]
A convex optimization problem is defined by two ingredients: [5] [6]
The goal of the problem is to find some attaining
In general, there are three options regarding the existence of a solution: [7] : chpt.4
A convex optimization problem is in standard form if it is written as
where: [7] : chpt.4
The feasible set of the optimization problem consists of all points satisfying the inequality and the equality 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. [7] : chpt.2
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. [8]
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: [9] : 1.4
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: [9] : 5.1
where K is a closed pointed convex cone, L is a linear subspace of Rn, and b is a vector in Rn. A linear program in standard form in the special case in which K is the nonnegative orthant of Rn.
It is possible to convert a convex program in standard form, to a convex program with no equality constraints. [7] : 132 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:
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.
The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations: [7] : chpt.4 [10]
Other special cases include;
The following are useful properties of convex optimization problems: [11] [7] : chpt.4
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 ]
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. [7] : chpt.11
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. [7] : chpt.11 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).
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. [7] : chpt.11 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. [7] : chpt.11
Convex optimization problems can also be solved by the following contemporary methods: [12]
Subgradient methods can be implemented simply and so are widely used. [15] 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 ]
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 .
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.
Program | Language | Description | FOSS? | Ref |
---|---|---|---|---|
CVX | MATLAB | Interfaces with SeDuMi and SDPT3 solvers; designed to only express convex optimization problems. | Yes | [16] |
CVXMOD | Python | Interfaces with the CVXOPT solver. | Yes | [16] |
CVXPY | Python | [17] | ||
Convex.jl | Julia | Disciplined convex programming, supports many solvers. | Yes | [18] |
CVXR | R | Yes | [19] | |
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 | [16] |
LMI lab | MATLAB | Expresses and solves semidefinite programming problems (called "linear matrix inequalities") | No | [16] |
LMIlab translator | Transforms LMI lab problems into SDP problems. | Yes | [16] | |
xLMI | MATLAB | Similar to LMI lab, but uses the SeDuMi solver. | Yes | [16] |
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 | [16] | |
ROME | Modeling system for robust optimization. Supports distributionally robust optimization and uncertainty sets. | Yes | [16] | |
GloptiPoly 3 | MATLAB, Octave | Modeling system for polynomial optimization. | Yes | [16] |
SOSTOOLS | Modeling system for polynomial optimization. Uses SDPT3 and SeDuMi. Requires Symbolic Computation Toolbox. | Yes | [16] | |
SparsePOP | Modeling system for polynomial optimization. Uses the SDPA or SeDuMi solvers. | Yes | [16] | |
CPLEX | Supports primal-dual methods for LP + SOCP. Can solve LP, QP, SOCP, and mixed integer linear programming problems. | No | [16] | |
CSDP | C | Supports primal-dual methods for LP + SDP. Interfaces available for MATLAB, R, and Python. Parallel version available. SDP solver. | Yes | [16] |
CVXOPT | Python | Supports primal-dual methods for LP + SOCP + SDP. Uses Nesterov-Todd scaling. Interfaces to MOSEK and DSDP. | Yes | [16] |
MOSEK | Supports primal-dual methods for LP + SOCP. | No | [16] | |
SeDuMi | MATLAB, Octave, MEX | Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP. | Yes | [16] |
SDPA | C++ | Solves LP + SDP. Supports primal-dual methods for LP + SDP. Parallelized and extended precision versions are available. | Yes | [16] |
SDPT3 | MATLAB, Octave, MEX | Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP. | Yes | [16] |
ConicBundle | Supports general-purpose codes for LP + SOCP + SDP. Uses a bundle method. Special support for SDP and SOCP constraints. | Yes | [16] | |
DSDP | Supports general-purpose codes for LP + SDP. Uses a dual interior point method. | Yes | [16] | |
LOQO | Supports general-purpose codes for SOCP, which it treats as a nonlinear programming problem. | No | [16] | |
PENNON | Supports general-purpose codes. Uses an augmented Lagrangian method, especially for problems with SDP constraints. | No | [16] | |
SDPLR | Supports general-purpose codes. Uses low-rank factorization with an augmented Lagrangian method. | Yes | [16] | |
GAMS | Modeling system for linear, nonlinear, mixed integer linear/nonlinear, and second-order cone programming problems. | No | [16] | |
Optimization Services | XML standard for encoding optimization problems and solutions. | [16] |
Convex optimization can be used to model problems in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, [7] : 17 data analysis and modeling, finance, statistics (optimal experimental design), [20] and structural optimization, where the approximation concept has proven to be efficient. [7] [21] Convex optimization can be used to model problems in the following fields:
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 ]
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In machine learning, support vector machines are supervised max-margin models with associated learning algorithms that analyze data for classification and regression analysis. Developed at AT&T Bell Laboratories by Vladimir Vapnik with colleagues SVMs are one of the most studied models, being based on statistical learning frameworks or VC theory proposed by Vapnik and Chervonenkis (1974).
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints. It is named after the mathematician Joseph-Louis Lagrange.
The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function. Since a sum of squares must be nonnegative, the algorithm can be viewed as using Newton's method to iteratively approximate zeroes of the components of the sum, and thus minimizing the sum. In this sense, the algorithm is also an effective method for solving overdetermined systems of equations. It has the advantage that second derivatives, which can be challenging to compute, are not required.
Interior-point methods are algorithms for solving linear and non-linear convex optimization problems. IPMs combine two advantages of previously-known algorithms:
In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations via a generalized secant method.
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function.
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem. Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality.
In mathematical optimization, constrained optimization is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized. Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied.
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function.
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
In the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem. A solution to the relaxed problem is an approximate solution to the original problem, and provides useful information.
Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization which may be considered a quasi-Newton method. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable, but not necessarily convex.
Subgradient methods are convex optimization methods which use subderivatives. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function. When the objective function is differentiable, sub-gradient methods for unconstrained problems use the same search direction as the method of steepest descent.
In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1.
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective, but the augmented Lagrangian method adds yet another term designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with, the method of Lagrange multipliers.
In mathematics, a submodular set function is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit. The natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory and electrical networks. Recently, submodular functions have also found utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains.
Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. Different optimization techniques are applied in various fields such as mechanics, economics and engineering, and as the complexity and amount of data involved rise, more efficient ways of solving optimization problems are needed. Quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.