ILOG CPLEX 11.0 User's Manual

subject to Ax ~ band a T i x + x T Q i x ≤ r i for i=1,...,qwith these bounds l ≤ x ≤ uAs with a quadratic objective function, convexity plays an important role in quadraticconstraints. The constraints must each define a convex region. To make sure of convexity,ILOG CPLEX requires that each Q i matrix be positive semi-definite (PSD) or that theconstraint must be in the form of a second order cone. The following sections offer moreinformation about these concepts.ConvexityThe inequality x 2 + y 2 ≤ 1 is convex. To give you an intuitive idea about convexity,Figure 13.1 graphs that inequality and shades the area that it defines as a constraint. If youconsider a and b as arbitrary values in the domain of the constraint, you see that anycontinuous line segment between them is contained entirely in the domain.Figure 13.1y(0, 1)(-1, 0)ab(1, 0)x(-1, -1)Figure 13.1 x 2 + y 2 ≤ 1 is convexThe inequality x 2 + y 2 ≥ 1 is not convex; it is concave. Figure 13.2 graphs that inequality andshades the area that it defines as a constraint. If you consider c and d as arbitrary values inthe domain of this constraint, then you see that there may be continuous line segments thatjoin the two values in the domain but pass outside the domain of the constraint to do so.240 ILOG CPLEX 11.0 — USER’ S MANUAL