v2010.10.26 - Convex Optimization

2.4. HALFSPACE, HYPERPLANE 832.4.2.6.2 Example. Minimization over hypercube.Consider minimization of a linear function over a hypercube, given vector cminimize c T xxsubject to −1 ≼ x ≼ 1(132)This convex optimization problem is called a linear program 2.23 because theobjective 2.24 of minimization c T x is a linear function of variable x and theconstraints describe a polyhedron (intersection of a finite number of halfspacesand hyperplanes).Any vector x satisfying the constraints is called a feasible solution.Applying graphical concepts from Figure 26, Figure 28, and Figure 29,x ⋆ = − sgn(c) is an optimal solution to this minimization problem but is notnecessarily unique. It generally holds for optimization problem solutions:optimal ⇒ feasible (133)Because an optimal solution always exists at a hypercube vertex (2.6.1.0.1)regardless of value of nonzero vector c in (132), [94] mathematicians see thisgeometry as a means to relax a discrete problem (whose desired solution isinteger or combinatorial, confer Example 4.2.3.1.1). [239,3.1] [240] 2.4.2.6.3 Exercise. Unbounded below.Suppose instead we minimize over the unit hypersphere in Example 2.4.2.6.2;‖x‖ ≤ 1. What is an expression for optimal solution now? Is that programstill linear?Now suppose minimization of absolute value in (132). Are the followingprograms equivalent for some arbitrary real convex set C ? (confer (517))minimize |x|x∈Rsubject to −1 ≤ x ≤ 1x ∈ C≡minimize α + βα , βsubject to 1 ≥ β ≥ 01 ≥ α ≥ 0α − β ∈ C(134)Many optimization problems of interest and some methods of solutionrequire nonnegative variables. The method illustrated below splits a variable2.23 The term program has its roots in economics. It was originally meant with regard toa plan or to efficient organization or systematization of some industrial process. [94,2]2.24 The objective is the function that is argument to minimization or maximization.