Convex Optimization 2026, 13 - Generalized Inequality Constraints
The standard form convex optimization problem can be generalized if the inequality constraint functions to be vector-valued, and using generalized inequalities in the constraints. If f0 is a proper cone, and fi are Ki-convex. This constitutes a standard form convex optimization problem with generalized inequality constraints (4.6.0). Among this, is the conic form problem, which aims to minimize cTx, and subject to Fx + g ⪯K 0; Ax = b. Picking either one of the constraint forms exclusively leads to the conic problem in standard form, or the conic form problem in inequality form (4.6.1). If K is 𝕊k+, conic form problems are semidefinite programs (SDP). This, too has a standard form with linear equality constraints and additionally a matrix nonnegativity constraint. The inequality form is defined analogously.
The standard form problem can be extended to include vector-valued constraint functions. If the objective function is also vector-valued, it extends to the general vector optimization problem (4.7.1). Generally, this extension only differs from the optimization. If the vector optimization problem is K-convex, the inequality constraint functions are convex and the equality constraints are affine. The optimal points of a vector optimization problem again form a set of feasible points. It can have a minimum element in that for some x f0(x) ⪯K f0(y) for all feasible y. This would make x optimal. A point x is optimal iff it's feasible and Ο ⊆ f0(x*) + K.
Given a the case where the set of achievable objective values don't have a minimum, the problem is expected not to have an optimal point. Minimal elements of the set of achievable values may still exist. A feasible point x is Pareto optimal if f0(x) is a minimal element of the set of achievable values. In this case, f0(x) is a Pareto optimal value for the vector optimization problem. Pareto optimality implies that for all feasible y, f0(y) ⪯K f0(x) ⇒ f0(y) = f0(x). Pareto optimality is equivalent to (f0(x) - K) ∩ Ο = {f0(x)}. Vector optimization problems can have several Pareto optimal values, the set of which is written as ℙ ⊆ Ο ∩ bd Ο where bd denotes the boundary operator (4.7.3). Finding the standard technique for finding Pareto optimal points for vector optimization problems can be done through scalarization of the problem (4.7.4.0). If the vector optimization problem is convex, then its scalarized problem is also convex. The Pareto-optimal points of a convex vector optimization problem can be determined by solving a convex scalar optimization problem. The weight vector determines the exact Pareto-optimal point (4.7.4.1, p1). The converse is partially true. It gives some nonzero weight vector for each Pareto-optimal point that is a solution of the scalarized problem (4.7.4.1. p2). This partial converse can be used occasionally to determine all Pareto optimal points of a convex vector optimization problem, in which case the Pareto optimal point is considered extreme.
Vector optimization problems involving the cone K = ℝq+, it's a multicriterion or multi-objective optimization problem. The components of f0 can be interpreted as q different scalar objectives, each is to be minimized. Fi is the ith objective of the problem. These need to be convex, along for fi being convex and hi being affine (4.7.5).