Gerstewitz Vectorization in Optimization

• Gerstewitz vectorization is a suite of techniques that uses Gerstewitz functionals to scalarize complex multiobjective and set-valued problems.
• It unifies methods like Pascoletti–Serafini scalarization and extends to nonconvex cases by mapping problems into a two-dimensional vector space.
• The approach leverages order structures and geometric properties to ensure tractable optimization, retrieving efficient solutions under compactness and stability conditions.

Gerstewitz vectorization refers to a suite of techniques in vector and set-valued optimization that enable scalarization or reduction of complex multiobjective and set-valued problems to tractable scalar or finite-dimensional vector optimization problems. The approach builds on Gerstewitz functionals, which provide a parametric way to represent efficient points, generalized orderings, and Pareto frontiers in ordered topological vector spaces. Two principal streams are established: the scalar Gerstewitz functional method in vector optimization, and the Gerstewitz vectorizing function for set-valued optimization, including nonconvex cases. The framework unifies and extends classical scalarization approaches, notably encompassing Pascoletti–Serafini scalarization as a special parameter choice.

1. Gerstewitz Functional: Definition and Properties

Let $Y$ be a real topological vector space. Given a closed reference set $A \subset Y$ and a nonzero direction $k \in -0^+A$ (where $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$ is the recession cone), the Gerstewitz functional is defined by

$$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$

Key properties include:

• Lower semicontinuity on $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$.
• Sublevel sets: For any $t \in \mathbb{R}$, the set $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$.
• Properness and finiteness on the effective domain if $A$ is convex and $k \in -\mathrm{core}\,0^+A$, and finiteness everywhere if $A \subset Y$0 (Weidner, 2017).

The geometric interpretation is that, for fixed $A \subset Y$1, $A \subset Y$2 is shifted along the line $A \subset Y$3 until it first “covers” $A \subset Y$4.

2. Scalarization in Vector and Set-Valued Optimization

In multicriteria optimization, the identification of efficient solutions (minima with respect to a cone $A \subset Y$5) is achieved by minimizing the Gerstewitz functional over the feasible image set: $A \subset Y$6 where $A \subset Y$7, $A \subset Y$8 is a reference point, and $A \subset Y$9 is a closed domination set (typically a cone). This scalarization is equivalent to the classical Pascoletti–Serafini scalar problem when $k \in -0^+A$0 and $k \in -0^+A$1 corresponds to the reference point (Weidner, 2017). All (weakly) efficient solutions of the original vector problem can be recovered via suitable parameterization.

For set-valued optimization (where the objective is a set-valued map $k \in -0^+A$2), the Gerstewitz vectorizing function enables reduction to a vector optimization problem in $k \in -0^+A$3. Given two sets $k \in -0^+A$4 and a direction $k \in -0^+A$5 for a closed convex pointed cone $k \in -0^+A$6,

$k \in -0^+A$7

where

$k \in -0^+A$8

and $k \in -0^+A$9 (Karaman et al., 2017).

3. Existence, Compactness, and Stability of Minimizers

The nonemptiness and compactness of the set of minimizers $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$0 depend on the interplay of the feasible set $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$1, the reference and ordering sets, and the direction:

• Necessity: $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$2 and $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$3 with $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$4.
• Sufficiency: If $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$5 such that $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$6 is nonempty and compact, then boundedness below of $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$7 on $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$8 is equivalent to finiteness on $0^+A := \{u\in Y\mid A+\mathbb{R}_+u\subset A\}$9; in this case, the minimizer set is nonempty and compact (Weidner, 2017).
• Compactness persists under various geometric conditions on $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$0 and $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$1. If $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$2 is compact and $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$3, compactness is guaranteed.

Parameter sensitivity:

• Scaling invariance: Replacing $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$4 by a positive multiple or shifting $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$5 along $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$6 leaves the minimizer set unchanged.
• Feasibility region convexity: If $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$7 is convex with non-trivial recession cone and $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$8 is convex, the feasible parameters $$\varphi_{A,k}(y) := \inf\{t \in \mathbb{R} \mid y \in A + t k\}, \qquad y \in Y.$$9 form a convex set (Weidner, 2017).

Stability:

• If $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$0 is compact for one parameter pair $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$1, compactness persists under perturbations $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$2.
• If minimizers are empty or unbounded for some $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$3, this persists for all nearby $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$4 outside $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$5.

4. Gerstewitz Vectorization for Nonconvex Set-Valued Problems

The Gerstewitz vectorizing function serves as a robust reduction for set-valued optimization, mapping a set-valued minimization problem to a two-dimensional vector optimization: $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$6 where minimization is with respect to the set-less order

$\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$7

for a closed convex cone $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$8 (Karaman et al., 2017).

Properties:

• Well-defined whenever $\operatorname{Dom}\varphi_{A,k} = A + \mathbb{R}k$9 are $t \in \mathbb{R}$0- and $t \in \mathbb{R}$1-bounded for all $t \in \mathbb{R}$2.
• The mapping $t \in \mathbb{R}$3 is invariant under the equivalence relation associated with $t \in \mathbb{R}$4, and strictly monotone under strict set-less ordering.
• The vectorizing function completely characterizes $t \in \mathbb{R}$5: $t \in \mathbb{R}$6.
• The approach is agnostic to convexity—solutions are preserved for nonconvex images, in contrast to scalarizations limited to convex settings (Karaman et al., 2017).

Optimality: Existence and characterization of $t \in \mathbb{R}$7-maximal (or minimal, weak) points are equivalent to Pareto maximality (resp., minimality) of $t \in \mathbb{R}$8 in $t \in \mathbb{R}$9.

5. Connections to Pascoletti–Serafini Scalarization and Generalizations

Pascoletti–Serafini scalarization parses efficient points of $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$0 by minimizing

$\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$1

with ordering cone $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$2, reference $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$3, and direction $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$4. Setting $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$5 and $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$6 in the Gerstewitz framework recovers this form precisely (Weidner, 2017).

The equivalence is explicit:

• Every Pascoletti–Serafini subproblem is a Gerstewitz minimization for suitable parameters.
• Conversely, any Gerstewitz scalarization $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$7 can be written in Pascoletti–Serafini form by matching $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$8 and $\{y \mid \varphi_{A,k}(y) \le t\} = A + t k$9, and setting the reference accordingly.
• The Gerstewitz approach allows parametrization with arbitrary closed $A$0, not limited to cones.

6. Implementation Guidelines and Limitations

Implementation of Gerstewitz vectorization and scalarization relies on careful parameter selection and geometry:

• The direction $A$1 should be in $A$2 for well-posedness and bounded, compact minimizer sets.
• Only one direction per “ray” is necessary; normalization by unit norm or sum-to-one convention is standard.
• Reference point $A$3 may be fixed on a hyperplane complementary to $A$4.
• For compact feasible sets $A$5, a grid over the boundary contour of $A$6 with fixed $A$7 suffices to recover the efficient frontier.
• If the minimizer set is empty or unbounded for some parameters, nearby choices will not resolve this unless $A$8 remains in $A$9.

Computational aspects in the set-valued, nonconvex setting (as with the Gerstewitz vectorizing function) reduce to searching for Pareto optima in $k \in -\mathrm{core}\,0^+A$0, which is algorithmically accessible, though explicit duality or specialized algorithms remain largely undeveloped (Karaman et al., 2017).

7. Illustrative Examples and Further Directions

Explicit examples illustrate the practical reduction:

• For $k \in -\mathrm{core}\,0^+A$1, $k \in -\mathrm{core}\,0^+A$2, $k \in -\mathrm{core}\,0^+A$3, $k \in -\mathrm{core}\,0^+A$4, and $k \in -\mathrm{core}\,0^+A$5, the unique minimizer is $k \in -\mathrm{core}\,0^+A$6, at $k \in -\mathrm{core}\,0^+A$7.
• In the nonconvex set-valued case, set-valued images consisting of ball unions or nonconvex compact sets are analyzed using the vectorizing function, yielding correct sets of minimal solutions, even when classical convexification fails (Karaman et al., 2017).

Limitations and open problems include the necessity for both $k \in -\mathrm{core}\,0^+A$8- and $k \in -\mathrm{core}\,0^+A$9-boundedness, dependence on the selection of $A \subset Y$00, restriction to two-dimensional surrogates, unresolved issues in differentiability and subdifferential calculus for $A \subset Y$01, and the lack of algorithmic development for more general settings, especially regarding higher-dimensional vectorizations or removal of comparability requirements (Karaman et al., 2017).

References: For detailed proofs, additional technical conditions, and extensive examples, see “Minimizers of Gerstewitz functionals” (Weidner, 2017) and “A Vectorization for Nonconvex Set-valued Optimization” (Karaman et al., 2017).