Mixed-Integer Multi-Objective Optimization Problem

• The mixed-integer multi-objective optimization framework combines continuous and integer variables to optimize conflicting objectives under complex algebraic constraints, with applications in energy and logistics.
• Scalarization methods, such as weighted-sum, ε-constraint, and Chebyshev techniques, are central to approximating nonconvex and disconnected Pareto fronts.
• Advanced algorithms, including hybrid metaheuristic and branch-and-bound approaches, enhance computational efficiency and scalability for large-scale, real-world problems.

A mixed-integer multi-objective optimization problem (MIMOP) is an optimization framework that simultaneously optimizes multiple conflicting objective functions over a feasible region defined by a combination of continuous and discrete (often binary or integer) decision variables, subject to complex algebraic constraints or implicit relations. This paradigm captures the essential combinatorial and tradeoff-rich structure of numerous real-world engineering, energy, logistics, and systems design tasks, especially those that require commitment logic, discrete investments, or switching decisions in addition to continuous resource allocations. Core methodological advances center on compact mathematical formulations, scalarization and frontier generation techniques, practical solver integration, and post-formulation modeling strategies that control computational tractability and interpretability.

1. Mathematical Structure and Formulation

A general MIMOP can be represented as follows: $$\begin{aligned} &\min (f_1(x),\,f_2(x),\dots,f_m(x)) \ &\text{s.t.}\quad x = (x_c, x_i) \in \mathbb{R}^{n_c} \times \mathbb{Z}^{n_i}, \ &\qquad g_j(x) \le 0, \quad j=1,\dots,p, \ &\qquad h_k(x) = 0, \quad k=1,\dots,q. \end{aligned}$$ Here, $x_c$ are continuous variables, $x_i$ are integer variables, $f_{\ell}$: objective functions (often linear or convex, but not restricted so), $g_j$, $h_k$: constraints (linear, nonlinear, or piecewise).

Each feasible $x$ yields an objective vector $f(x)$; a point is called (weakly) Pareto-optimal if no other feasible $x'$ satisfies $f_\ell(x') \le f_\ell(x)$ for all $\ell$ with strict inequality for at least one. The collection of these vectors is the Pareto front, which is typically disconnected and nonconvex in the mixed-integer case (Burachik et al., 2019, Przybylski et al., 2019).

A canonical instance in the literature is the Mixed-Integer Tri-Objective Unit Commitment (MIP-TUC) for energy systems:

• $$\{x_{t,v}^{\mathrm{in},r}, x_{t,v}^{\mathrm{out},r}\}$$: flow variables (continuous)
• $\{z_{i,t}, s_{i,t}\}$: unit on/off status, start-up (binary)
• $h_{t,k}^r$: storage (continuous)
• $p_t^r, e_t^r$: purchases/sales (continuous) with cost, emissions, and output as objectives, and time-coupled technical constraints for flows, conversions, ramping, storage, balance, and logical relationships (Riedmüller et al., 20 May 2025).

2. Scalarization and Pareto-Front Generation Techniques

Direct computation of the full Pareto front in MIMOPs is infeasible for all but trivial cases; practical solution methods revolve around scalarization—transforming the multi-objective problem to a parametrized family of single-objective subproblems.

Classical scalarization approaches (Burachik et al., 2019, Przybylski et al., 2019, Tevruez et al., 13 Jan 2025):

• Weighted-Sum: Minimize $w^\top f(x)$ for $w \in \mathbb{R}^m_+$, $\sum w_i=1$. Supported Pareto points (front's convex hull) are recovered, but nonconvex or disconnected front sections are missed.
• $\epsilon$-Constraint: Minimize $f_k(x)$ subject to $f_\ell(x) \le \epsilon_\ell$ for $\ell \neq k$. Can recover nonconvex regions but requires careful selection of $\epsilon$.
• Tchebycheff (Chebyshev): Minimize $\max_i w_i(f_i(x)-u_i)$ with $u$ a reference point. Captures both convex and nonconvex regions and is widely used for large-scale problems (Filcek et al., 2023).
• Weighted-Constraint Scalarization: For $m$ objectives, solve $m$ subproblems with weighted constraints and report non-dominated outcomes; robust to domain disconnection for $m \geq 3$ (Burachik et al., 2019).

Iterative and adaptive procedures:

• Dichotomic/Facet-based search: Alternate between convex hull identification in objective space and new scalarization solves to incrementally enumerate all extreme (supported) nondominated outcomes (Przybylski et al., 2019).
• Outer Approximation (OA) and Dual Benson: Construct and iteratively refine outer approximations (polyhedral hulls) of the feasible outcome set, requiring only access to a weighted-sum MIP (or MINLP) oracle (Bökler et al., 2021, Bökler et al., 2022).
• Adaptive-weight and segment refinement: Dynamically allocate scalarization weights where the front is under-resolved (e.g., via Euclidean segment length), increasing focus on sparse regions (Tevruez et al., 13 Jan 2025).

For nonlinear cases, algorithms such as MCBB+B&B combined with metaheuristics (e.g., NSGA-II) are employed to both approximate upper (incumbent) and lower bound sets, with pruning guided by Pareto dominance (Jaber et al., 2020).

3. Modeling, Formulation Decisions, and Post-Formulation Design

Significant degrees of freedom exist in how a real-world system is abstracted to an MIMOP, especially in network-based problems (energy, infrastructure). In (Riedmüller et al., 20 May 2025), the authors highlight two modeling poles:

• Model A—node-based, "human-readable": Hierarchical asset graphs, explicit nodes for subcomponents and information transport (cost, emission), node-attached flow variables. Advantages: transparency, domain interpretability, easy embedding of technology-specific constraints. Disadvantages: large raw MIP size due to variable duplication and redundant constraints.
• Model B—arc-based, "mathematical flexibility": Lean, homogeneous graphs of only real physical flows, arc-focused variables, generic constraint embedding. Advantages: smaller graph and MIP, suited to generic solvers, minimal memory footprint. Disadvantages: reduced interpretability, cross-node constraint entanglement.

These choices do not alter the feasible set or front up to contraction equivalence but can alter solver performance via presolve efficiency, data handling, and ease of post-hoc analysis. The main recommendation is to balance model clarity with computational compactness according to system size and intended user interaction (Riedmüller et al., 20 May 2025).

4. Algorithmic Approaches and Solvers

Diverse algorithmic strategies are implemented in both exact and metaheuristic frameworks:

• Branch-and-Bound (B&B) plus Metaheuristics: MCBB tree search combined with population-based evolutionary algorithms such as NSGA-II at root, intermediate, or leaf nodes. Fathoming with lower (relaxed) and upper (incumbent) Pareto bound sets (Jaber et al., 2020).
• Hybrid Heuristic–Exact Integration: Use of NSGA-II to explore feasible space and guide B&B fathoming, or hybridization with surrogate models (e.g., GP, Kriging) and robust dominance checks.
• Interval Representations for Incomplete or Large-Scale Problems: For large MOMIPs where single scalarization solves are too expensive, methods provide componentwise interval bounds $[L_\ell, U_\ell]$ for Pareto-representative solutions obtained within time or gap limits, enabling approximate navigability of the front (Filcek et al., 2023).
• Dedicated Software (PaMILO): Implements output-sensitive Dual Benson algorithm for MOMILPs and MOMIQCQPs, leveraging industrial MIP solvers and polyhedral libraries, and producing the full set of extreme Pareto points and their supporting facets (Bökler et al., 2022).
• Metaheuristics for Black-Box and Nonlinear Problems: Modified NSGA-II, multi-objective CMA-ES with marginal integer-handling, and linesearch-based direct search for derivative-free or nonsmooth MIMOPs (Khavar et al., 25 Jul 2025, Hamano et al., 2022, Liuzzi et al., 2022).

Practical guidance includes warm-starting solvers over adjacent weights, dynamic constraint filtration, adaptive parameter choices, parallelization over independently-solvable scalarizations or B&B nodes, and custom crowding- or diversity-metrics to ensure uniform Pareto coverage.

5. Benchmarks, Applications, and Computational Insights

MIMOPs are motivated by applications in energy system operation (unit commitment, district heating, distribution automation), industrial design (beam, vessel, and gear optimization), statistical learning (best-subset regression), and black-box or simulation-based settings (material design, surrogate modeling in MixMOBO) (Riedmüller et al., 20 May 2025, Khavar et al., 25 Jul 2025, Charkhgard et al., 2018, Sheikh et al., 2022).

Computational studies consistently demonstrate the following:

Approach	Problem/Front Size	Characteristics	Observed Trade-offs
Node-based MIP	128 nodes/74k vars	Readable, large raw MI	Fast presolve, user-friendly
Arc-based MIP	33 nodes/16.7k vars	Minimal raw MI	Fast presolve, lower RAM, less clear
Adaptive weights	100 Pareto points	Automated segment ref.	Efficient frontier filling
MCBB+B&B+NSGA-II	20-30 variables	Mixed discrete/continuous	True Pareto set for discrete, robust

Notable observations include the ability of minimal arc-based models to yield substantial raw size reductions (about 75%) prior to MIP solver presolve, with nearly identical solution-phase metrics post-presolve (Riedmüller et al., 20 May 2025), and the effectiveness of advanced hybrid algorithms in achieving high-fidelity Pareto approximation with acceptable computational investment (Jaber et al., 2020). Interval-based Pareto representations provide guaranteed gap levels on large instances at fixed time limits, with typical suboptimality gaps $\approx 1-5\%$ of shell bounds for two/three-objective knapsack problems (Filcek et al., 2023).

6. Challenges and Future Directions

Major open and emerging issues for MIMOPs include:

• Disconnected and nonconvex Pareto fronts: Variable space discontinuities require specialized weighted-constraint scalarizations and front construction techniques (Burachik et al., 2019).
• Balancing model clarity and memory/performance: Excessively compact MIPs obscure interpretation, while redundant node-based formulations inflate size.
• Large-scale and real-time use: Handling time-constrained solution limits via interval representation or relaxation hierarchy (Filcek et al., 2023).
• Integration of advanced relaxations: Use of McCormick envelopes, Lagrangian or conic relaxations for handling bilinear, quadratic, or otherwise nonconvex terms (Tevruez et al., 13 Jan 2025).
• Metaheuristic handling for black-box, nonsmooth, or simulation-derived objectives: Adaptation of marginal-probability mechanisms and hybrid CMA-ES/NSGA-II variants enhances robustness but prompts further paper in categorical/empirical settings (Hamano et al., 2022, Sheikh et al., 2022).
• Parallelization and warm-starting: Critical for both exact and metaheuristic frameworks given the explosion in subproblem count as the number of objectives or variables increases (Bökler et al., 2021, Bökler et al., 2022).

7. Summary Table: Key Scalarization and Algorithmic Approaches

Technique	Front Coverage	Supported/Unsupported	Scaling Properties	Cited Works
Weighted-Sum	Convex only	Supported only	Efficient for small $m$	(Burachik et al., 2019)
$\epsilon$-Constraint	Convex + nonconvex	Some unsupported	Grid grows exponentially in $m$	(Burachik et al., 2019, Tevruez et al., 13 Jan 2025)
Chebyshev	Full (with grid)	Supported+some unsupported	Tight, good for large-scale	(Filcek et al., 2023)
Weighted-Constraint	Disconnected, nonconvex	All, via aggregation	$m$-fold cost	(Burachik et al., 2019)
Dichotomic/Facet Search	All extremes	Supported	Output-sensitive, poly. in points	(Przybylski et al., 2019)
Outer Approx./Dual-Benson	All extremes/facets	Supported	Output-sensitive, poly. per call	(Bökler et al., 2021, Bökler et al., 2022)
Hybrid Metaheuristic+B&B	Approximated front	Both, for nonconvex	Can reach true discrete Pareto, scalable	(Jaber et al., 2020)

Each approach is chosen according to problem structure, front geometry, solver availability, and the desired balance between front fidelity and computational cost. Cross-integration and algorithmic hybridization, including local exploration with evolutionary search and global pruning or exact hull enumeration, are increasingly prevalent in state-of-the-art MIMOP toolkits.