Mond-Weir Dual-based Program (MDP)

• Mond-Weir Dual-based Program is a single-level reformulation for bilevel optimization that replaces traditional KKT conditions with the Mond-Weir dual to handle multiobjective, nonlinear, non-convex, and nonsmooth problems.
• It exhibits both weak and strong duality under mild generalized convexity and standard constraint qualifications, ensuring reliable equivalence between the original and reformulated problems.
• Algorithmic implementations using direct projection and relaxation methods demonstrate MDP's practical efficiency and robustness compared to traditional reformulations like MPCC.

The Mond-Weir Dual-based Program (MDP) is a single-level reformulation approach for bilevel optimization problems, grounded in the use of the Mond-Weir (MW) dual of the lower-level problem instead of traditional Karush-Kuhn-Tucker (KKT) or Wolfe dual-based methods. The MDP is designed to achieve theoretical and practical advantages for broad classes of bilevel programs, including multiobjective, nonlinear, non-convex, and nonsmooth variants. It is characterized by weaker complementarity conditions than classical approaches, favorable constraint qualifications, and strong duality properties under mild regularity and generalized convexity assumptions.

1. Mathematical Formulation and Construction

For a standard bilevel program, the upper level is: $$\min_{x \in \mathbb{R}^n,\; y \in \mathbb{R}^m} F(x, y) \quad \text{s.t.} \quad (x, y) \in \Omega,\quad y \in S(x)$$ where $$S(x) = \arg\min_{y \in \mathbb{R}^m} \{f(x, y) \mid g(x,y) \leq 0,\, h(x,y) = 0 \}$$ defines the lower-level feasible set. The associated lower-level Lagrangian is: $$L(x, z, u, v) = f(x, z) + u^T g(x, z) + v^T h(x, z)$$ The Mond-Weir dual of the lower-level problem is: $$(\mathrm{MD}_x)\quad \max_{z,u,v}\; f(x,z) \quad \text{s.t.}\; \begin{cases} \nabla_z L(x,z,u,v) = 0\ g(x,z) \le 0, \quad h(x,z) = 0\ u \ge 0,\quad u^T g(x, z) \ge 0 \end{cases}$$

The MDP single-level reformulation replaces the lower-level optimality condition $y \in S(x)$ with the MW dual feasibility conditions and an upper-level linking constraint: $\min_{x, y, z, u, v} F(x, y)$

$$\text{s.t.}\; \begin{cases} (x, y) \in \Omega,\ g(x, y) \leq 0,\quad h(x, y) = 0,\ \nabla_z L(x, z, u, v) = 0,\ u \ge 0,\ u^T g(x, z) + v^T h(x, z) \ge 0,\ f(x, y) \le f(x, z) \end{cases}$$

This formulation jointly enforces (i) upper-level feasibility, (ii) that $y$ is no worse for the lower-level objective than the dual trial point $z$, and (iii) that $(z, u, v)$ satisfy MW dual feasibility and complementarity.

In the multiobjective and nonsmooth context (fractional multiobjective bilevels), directional convexificators are used to define (generalized) gradients in the absence of classical differentiability, enabling MDP construction for highly irregular problems (Lara et al., 22 Nov 2025).

2. Theoretical Properties: Duality and Equivalence

MDP inherits strong duality properties from the Mond-Weir dual under mild generalized convexity and constraint qualification assumptions:

• Weak Duality: For any feasible upper-level $(x, y)$ and MW dual feasible $(z, u, v)$, $f(x, y) \ge f(x, z)$ holds, provided $f(x, \cdot)$ is pseudoconvex and $u^T g(x, \cdot) + v^T h(x, \cdot)$ is quasiconvex (Li et al., 2023, Li et al., 3 Dec 2025).
• Strong Duality: If a Guignard-type constraint qualification (GCQ) holds at some lower-level solution $y^*$, then $$\min_{y\in Y(x)} f(x, y) = \max_{(z, u, v)\in M(x)} f(x, z)$$. This allows reformulation of the bilevel problem as a single-level optimization using the MDP constraints (Li et al., 3 Dec 2025).
• Global and Local Equivalence: Any global (resp. local) minimizer of the original bilevel program projects from a global (resp. local) minimizer of the MDP formulation, and vice versa, under the above assumptions (Li et al., 2023, Li et al., 3 Dec 2025).

In the multiobjective fractional context, the weak and strong duality theorems are extended using vector-valued value functions and set-inclusion/gradient conditions formulated in terms of directional convexificators, leveraging generalized convexity concepts such as pseudoconvexity and quasiconvexity (Lara et al., 22 Nov 2025).

3. Constraint Qualifications and Regularity

A critical property distinguishing MDP from classical KKT-based reformulations (MPCC) is its compatibility with standard constraint qualifications:

• Mangasarian-Fromovitz Constraint Qualification (MFCQ): MDP can satisfy MFCQ at feasible points unless degenerate cases (e.g., $z = y$) arise (Li et al., 2023, Li et al., 3 Dec 2025). In contrast, MPCC fails MFCQ at all feasible points, limiting solver efficiency and reliability.
• Guignard CQ: For strong duality and global equivalence, it suffices that the GCQ holds at some minimizer of the lower-level problem, a relatively mild requirement compared to classical constraint qualifications.

In nonsmooth settings, a generalized nonsmooth Abadie-type constraint qualification based on $\partial_D$-convexificators is used, broadening the class of problems addressable by the MDP methodology (Lara et al., 22 Nov 2025).

4. Algorithmic Implementation and Computational Methodology

MDP admits several algorithmic frameworks, with two predominant strategies:

• Direct Algorithm with Projection: Initialize decision variables, solve the MDP using nonlinear programming methods (e.g., SQP), and, if the upper-level feasibility or optimality gap is significant, project iterates and re-solve as necessary.
• Relaxation Algorithm: Introduce a relaxation parameter $t > 0$ in the linking condition $f(x, y) \le f(x, z) + t$, solve the relaxed problem, then decrease $t$ iteratively, warm-starting at each stage. This approach improves convergence for challenging instances (Li et al., 2023, Li et al., 3 Dec 2025).

For large-scale testbeds, both direct and relaxation algorithms have been shown to effectively navigate the feasible region, with relaxation methods yielding stronger numerical performance and global convergence guarantees under CPLD (constant positive linear dependence) assumptions (Li et al., 2023, Li et al., 3 Dec 2025).

5. Comparative Assessment with Alternative Reformulations

MDP has been systematically compared to other major single-level reformulations, notably MPCC (KKT-based) and WDP (Wolfe dual-based):

• MPCC Reformulation: Encounters severe failure of MFCQ and often performs poorly in feasibility and robustness tests (Li et al., 2023).
• WDP: Incorporates a single "Wolfe-constraint" and generally outperforms MPCC, but MDP often yields a smaller feasible region and can satisfy MFCQ at more feasible points.
• eMDP (Extended MDP): Further relaxes duality constraints, leading to increased numerical feasibility at the possible expense of tightness (Li et al., 2023).

This comparison is reflected in comprehensive computational studies summarized in the following table for relaxation algorithms on 450 random bilevel problems (Li et al., 3 Dec 2025):

Reformulation	Feasibility/Dominance Ratio vs MPCC	Avg. CPU Time (s, Relax.)
MDP	~4.77× (dominant solutions)	171
WDP	~4.72×	comparable
eMDP	~2.10×	variable
MPCC	Baseline	430

MDP outperforms MPCC and is competitive with or superior to WDP in both practical robustness and solution quality (Li et al., 2023, Li et al., 3 Dec 2025).

6. Extensions to Nonsmooth and Multiobjective Problems

MDP generalizes naturally to multiobjective and nonsmooth bilevel problems, especially with fractional objectives. In this context, the MDP incorporates directional convexificators—set-valued maps governing upper Dini derivatives and encapsulating generalized subdifferentials. This allows the formulation and proof of optimality conditions and duality results under minimal smoothness and differentiability requirements (Lara et al., 22 Nov 2025).

An illustrative example from (Lara et al., 22 Nov 2025) demonstrates the construction of a Mond-Weir dual (denoted (MQ)) for a two-objective fractional bilevel, verifies the existence of the required convexificators, and identifies feasible multipliers so that all MDP conditions are satisfied.

7. Limitations and Practical Considerations

The principal advantages of MDP are its capacity to:

• Handle multiobjective, nonsmooth, and discontinuous data;
• Satisfy standard constraint qualification assumptions at feasible points;
• Achieve both weak and strong duality under mild, generalized convexity;
• Yield practical numerical advantages in solution feasibility and efficiency relative to traditional reformulations (Li et al., 2023, Li et al., 3 Dec 2025).

However, limitations include:

• The requirement for heavy notation and technical apparatus (directional convexificators and underlying continuity cones);
• Potential challenges in verifying boundedness and upper-semi-regularity of relevant convexificators for nonsmooth models;
• Implementation effort needed to evaluate or approximate set-valued convexificators in practical algorithms;
• Subtlety of the generalized constraint qualifications (e.g., $\partial_D$-ACQ), which may fail in degenerate scenarios (Lara et al., 22 Nov 2025).

Overall, the Mond-Weir dual-based program provides a flexible, theoretically robust single-level bilevel reformulation framework with demonstrably strong computational performance and broad applicability across nonlinear, multiobjective, and nonsmooth hierarchical optimization problems.