Mathematical optimization Shapley–Folkman lemma

a function convex if region above graph convex set.

the shapley–folkman lemma has been used explain why large minimization problems non-convexities can solved (with iterative methods convergence proofs stated convex problems). shapley–folkman lemma has encouraged use of methods of convex minimization on other applications sums of many functions.

preliminaries of optimization theory

nonlinear optimization relies on following definitions functions:

the graph of function f set of pairs of arguments x , function evaluations f(x)


graph(f) = { (x, f(x) ) }


the epigraph of real-valued function f set of points above graph


the sine function non-convex.



epi(f) = { (x, u) : f(x) ≤ u }.


a real-valued function defined convex function if epigraph convex set.

for example, quadratic function f(x) = x convex, absolute value function g(x) = |x|. however, sine function (pictured) non-convex on interval (0, π).

additive optimization problems

in many optimization problems, objective function f separable: is, f sum of many summand-functions, each of has own argument:

f(x) = f( (x1, ..., xn) ) = ∑ fn(xn).

for example, problems of linear optimization separable. given separable problem optimal solution, fix optimal solution

xmin = (x1, ..., xn)min

with minimum value f(xmin). separable problem, consider optimal solution (xmin, f(xmin) ) convexified problem , convex hulls taken of graphs of summand functions. such optimal solution limit of sequence of points in convexified problem

(xj, f(xj) ) ∈  ∑ conv (graph( fn ) ).

of course, given optimal-point sum of points in graphs of original summands , of small number of convexified summands, shapley–folkman lemma.

this analysis published ivar ekeland in 1974 explain apparent convexity of separable problems many summands, despite non-convexity of summand problems. in 1973, young mathematician claude lemaréchal surprised success convex minimization methods on problems known non-convex; minimizing nonlinear problems, solution of dual problem problem need not provide useful information solving primal problem, unless primal problem convex , satisfy constraint qualification. lemaréchal s problem additively separable, , each summand function non-convex; nonetheless, solution dual problem provided close approximation primal problem s optimal value. ekeland s analysis explained success of methods of convex minimization on large , separable problems, despite non-convexities of summand functions. ekeland , later authors argued additive separability produced approximately convex aggregate problem, though summand functions non-convex. crucial step in these publications use of shapley–folkman lemma. shapley–folkman lemma has encouraged use of methods of convex minimization on other applications sums of many functions.