Convex optimization
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Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. The convexity makes optimization easier than the general case since local minimum must be a global minimum, and firstorder conditions are sufficient conditions for optimality.^{[1]}
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,^{[2]} data analysis and modeling, finance, statistics (optimal experimental design),^{[3]} and structural optimization.^{[4]} 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 reformulated equivalently as a problem of minimizing the function f, which is convex.
Contents
Definition
Given a real vector space together with a convex, realvalued function defined on a convex subset of
the problem is to find any point in for which the number is smallest, i.e., a point such that
 for all .
The convexity of makes the powerful tools of convex analysis applicable. In finitedimensional 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 optimization problem
The general form of an optimization problem (also referred to as a mathematical programming problem or minimization problem) is to find some such that
for some feasible set and objective function . The optimization problem is called a convex optimization problem if is a convex set and is a convex function defined on . ^{[5]} ^{[6]}
Alternatively, an optimization problem of the form
is called convex if the functions are all convex functions.^{[7]}
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 to be minimized over the variable
 Inequality constraints of the form , where the functions are convex
 Equality constraints of the form , where the functions are affine. In practice, the terms "linear" and "affine" are often used interchangeably. Such constraints can be expressed in the form , where is a columnvector, ( indicates Transpose), and a real number.
A convex minimization problem is thus written as
Note that every equality constraint can be equivalently replaced by a pair of inequality constraints and . 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 has to be affine as opposed to merely being convex. If is convex, is convex, but is concave. Therefore, the only way for to be convex is for to be affine.
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.
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
 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 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:
 minimizes over all
 with at least one
 (complementary slackness).
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 .
Methods
Convex minimization problems can be solved by the following contemporary methods:^{[8]}
 "Bundle methods" (Wolfe, Lemaréchal, Kiwiel), and
 Subgradient projection methods (Polyak),
 Interiorpoint methods (Nemirovskii and Nesterov).
Other methods of interest:
 Cuttingplane methods
 Ellipsoid method
 Subgradient method
 Dual subgradients and the driftpluspenalty method
Subgradient methods can be implemented simply and so are widely used.^{[9]} Dual subgradient methods are subgradient methods applied to a dual problem. The driftpluspenalty method is similar to the dual subgradient method, but takes a time average of the primal variables.
Convex minimization with good complexity: Selfconcordant 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;^{[10]} 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 selfconcordant barrier functions, according to the theory of Nesterov and Nemirovskii, and second selfregular barrier functions according to the theory of Terlaky and coauthors.
Quasiconvex minimization
Problems with convex level sets can be efficiently minimized, in theory. Yurii Nesterov proved that quasiconvex minimization problems could be solved efficiently, and his results were extended by Kiwiel.^{[11]} However, such theoretically "efficient" methods use "divergentseries" stepsize rules, which were first developed for classical subgradient methods. Classical subgradient methods using divergentseries 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 closetoconvex but nonconvex problems can be computationally intractable. Minimizing a unimodal function is intractable, regardless of the smoothness of the function, according to results of Ivanov.^{[12]}
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.^{[13]}
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.^{[14]} 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 NPhard.^{[15]} In fact, in the quadratic minimization problem, if the matrix has only one negative eigenvalue, is NPhard.^{[16]}
Extensions
Advanced treatments consider convex functions that can attain positive infinity, also; the indicator function of convex analysis is zero for every and positive infinity otherwise.
Extensions of convex functions include biconvex, pseudoconvex, and quasiconvex functions. Partial extensions of the theory of convex analysis and iterative methods for approximately solving nonconvex minimization problems occur in the field of generalized convexity ("abstract convex analysis").
See also
Notes
 ^ Rockafellar, R. Tyrrell (1993). "Lagrange multipliers and optimality" (PDF). SIAM Review. 35 (2): 183–238. doi:10.1137/1035044.
 ^ Boyd/Vandenberghe, p. 17.
 ^ Chritensen/Klarbring, chapter 4.
 ^ Boyd/Vandenberghe, chapter 7.
 ^ HiriartUrruty, JeanBaptiste; Lemaréchal, Claude (1996). Convex analysis and minimization algorithms: Fundamentals. p. 291.
 ^ BenTal, Aharon; Nemirovskiĭ, Arkadiĭ Semenovich (2001). Lectures on modern convex optimization: analysis, algorithms, and engineering applications. pp. 335–336.
 ^ Boyd/Vandenberghe, p. 7
 ^ For methods for convex minimization, see the volumes by HiriartUrruty and Lemaréchal (bundle) and the textbooks by Ruszczyński, Bertsekas, and Boyd and Vandenberghe (interior point).
 ^ Bertsekas
 ^ HiriartUrruty & Lemaréchal (1993, Example XV.1.1.2, p. 277) discuss a "bad guy" constructed by Arkadi Nemirovskii.

^ In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated):
Kiwiel, Krzysztof C. (2001). "Convergence and efficiency of subgradient methods for quasiconvex minimization". Mathematical Programming (Series A). 90 (1). Berlin, Heidelberg: Springer. pp. 1–25. doi:10.1007/PL00011414. ISSN 00255610. MR 1819784. Kiwiel acknowledges that Yurii Nesterov first established that quasiconvex minimization problems can be solved efficiently.
 ^ Nemirovskii and Judin
 ^ Convex maximization is mentioned in the subsection on convex optimization in this textbook: Ulrich Faigle, Walter Kern, and George Still. Algorithmic principles of mathematical programming. SpringerVerlag. Texts in Mathematics. Chapter 10.2, Subsection "Convex optimization", pages 205206.
 ^ Theorem 32.1 in Rockafellar's Convex Analysis states this maximum principle for extended realvalued functions.
 ^ Sahni, S. "Computationally related problems," in SIAM Journal on Computing, 3, 262279, 1974.
 ^ Quadratic programming with one negative eigenvalue is NPhard, Panos M. Pardalos and Stephen A. Vavasis in Journal of Global Optimization, Volume 1, Number 1, 1991, pg.1522.
References
 Bertsekas, Dimitri P.; Nedic, Angelia; Ozdaglar, Asuman (2003). Convex Analysis and Optimization. Belmont, MA.: Athena Scientific. ISBN 1886529450.
 Bertsekas, Dimitri P. (2009). Convex Optimization Theory. Belmont, MA.: Athena Scientific. ISBN 9781886529311.
 Bertsekas, Dimitri P. (2015). Convex Optimization Algorithms. Belmont, MA.: Athena Scientific. ISBN 9781886529281.
 Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 9780521833783. Retrieved October 15, 2011.
 Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
 HiriartUrruty, JeanBaptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
 HiriartUrruty, JeanBaptiste; Lemaréchal, Claude (1993). Convex analysis and minimization algorithms, Volume I: Fundamentals. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 305. Berlin: SpringerVerlag. pp. xviii+417. ISBN 3540568506. MR 1261420.
 HiriartUrruty, JeanBaptiste; Lemaréchal, Claude (1993). Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: SpringerVerlag. pp. xviii+346. ISBN 3540568522. MR 1295240.
 Kiwiel, Krzysztof C. (1985). Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. New York: SpringerVerlag. ISBN 9783540156420.
 Lemaréchal, Claude (2001). "Lagrangian relaxation". In Michael Jünger and Denis Naddef. Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000. Lecture Notes in Computer Science. 2241. Berlin: SpringerVerlag. pp. 112–156. doi:10.1007/3540455868_4. ISBN 3540428771. MR 1900016.
 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
 Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press.
 Ruszczyński, Andrzej (2006). Nonlinear Optimization. Princeton University Press.
 Christensen, Peter W.; Anders Klarbring (2008). An introduction to structural optimization. 153. Springer Science & Businees Media.
External links
Wikimedia Commons has media related to Convex optimization. 
 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