Mixed-Integer Semidefinite Programming

Updated 10 November 2025

Mixed-Integer Semidefinite Programming is an optimization paradigm that generalizes MILP and SDP by integrating matrix positive semidefiniteness with integer constraints to model combinatorial structures.
MISDP formulations yield compact and strong relaxations, enabling precise modeling of complex problems such as quadratic portfolio selection, max-cut, and graph partitioning.
Advances in algorithms, decomposition techniques, and dual bounding strategies have enhanced the scalability and robustness of MISDP for applications in control, robust optimization, and network design.

Mixed-Integer Semidefinite Programming (MISDP) is the class of optimization problems where some decision variables are restricted to integer values and the feasible set is defined by linear equalities, inequalities, and matrix positive semidefiniteness constraints. MISDP generalizes both mixed-integer linear programming (MILP) and semidefinite programming (SDP), and provides a powerful modeling paradigm for large classes of combinatorial, control, and robust optimization problems where structure can be captured via convex semidefinite constraints and integrality. MISDP formulations can exactly encode various discrete optimization problems, often yielding very compact or theoretically strong relaxations, and support the development of decomposition, bounding, and heuristic methods.

1. Canonical Formulation and Theoretical Foundation

A generic MISDP in block-structured form is given by: $\begin{aligned} \min\quad &\langle C,\,X\rangle \ \text{s.t.}\quad &\langle A_i,\,X\rangle = b_i,\quad i=1,\dots,m,\ &X\succeq 0,\ &X_{jj}\in\{0,1\}\ \forall\,j\in J,\quad X_{uv}\in\mathbb Z\ \forall(u,v) \in K, \end{aligned}$ where $X \in \mathcal{S}^n$ (symmetric matrices), $C, A_i \in \mathcal{S}^n$ , and integrality is enforced on specified entries. This form generalizes QCQPs, MILPs, and allows direct modeling of 0–1, $\pm1$ , and general integer-valued combinatorial structures (Meijer et al., 2023).

A striking structural result is that for binary PSD matrices, $X \in \{0,1\}^{n\times n}, X = X^\top, X \succeq 0$ , one always has a rank factorization $X = \sum_{j=1}^r x_j x_j^\top$ , $x_j \in \{0,1\}^n$ , with up to $\operatorname{rank}(X)$ terms. For rank-1, the following holds: $Y = \begin{pmatrix} 1 & x^\top \ x & X \end{pmatrix} \succeq 0,\ \operatorname{diag}(X)=x \implies \operatorname{rank}(Y)=1 \iff X=xx^\top,\ x\in\{0,1\}^n.$ This exactness enables strong formulations for many binary QCQPs and is central to the modeling power of MISDP (Meijer et al., 2023).

2. Reformulation and Relaxation Techniques

MISDPs typically encode nonconvexities through:

Integrality on matrix entries or auxiliary discrete variables
Nonlinear/nonconvex quadratic constraints via matrix lifting, e.g., $X = xx^\top$

Relaxations are constructed by dropping rank/integrality or using convexifications. For example, the practical approach to cardinality-constrained quadratic portfolio selection (Wiegele et al., 2021):

Original MIQP with selection variables
Lift to MISDP with semidefinite and rank-1 constraints
SDP relaxation by relaxing binary requirements, yielding a convex problem over PSD matrices This relaxation is extremely tight empirically, achieving rank-1 solutions in 96% of tested portfolio instances.

MISDPs underpin tight convexifications in enhanced relaxations for MIQCQPs (e.g., AC power flow with binary investment or switching):

Use enhanced linear equalities to strengthen SDP relaxations (Li, 2015)
Reformulate nonconvexities using disjunctive programming and convex-hull descriptions
The resulting “convex hull” MISDP is solved efficiently via branch-and-bound, where each node’s continuous relaxation is the convex hull of a local disjunction, preserving theoretical tightness

3. Algorithms and Decomposition Paradigms

Solving MISDPs is computationally intensive (NP-hard), motivating specialized algorithms:

Branch-and-Bound: Branch on integrality, solve SDP relaxations at nodes; tight SDP relaxations greatly reduce tree size (Wiegele et al., 2021, Liu et al., 2023, Li, 2015).
Decomposition: Decompose large MISDPs into MIQP-masters and SDP-subproblems, passing coupling cuts (no-good, non-revenue-power, Lagrangian cuts). Example: Transmission-constrained unit commitment (Gambella et al., 2018), guaranteeing convergence in finitely many steps by exhaustive enumeration and cut generation.
First-order Dual Methods: Lagrangian duality for MISDPs yields dual bounds always at least as strong as the continuous SDP relaxation. Hierarchies of dual bounds are available by partially enforcing integrality/PSD on small submatrices (“packings”), solvable by projected subgradient, accelerated subgradient, or proximal bundle methods (Meijer et al., 20 Jan 2025).
Heuristic and Penalization: Difference-of-convex decompositions and convex–concave procedures allow for practical solution of large MISDPs with complementarity or logical constraints, as in robust topology optimization (Kanno, 2017).

Dual Level	% Gap Closed	Solve Time (s) (n=100)
SDP relaxation	0	<1
LD, p=3 (triples)	38–55	1–6
LD, p=7 (septs)	45–57	up to 15

These results document that even moderate dual hierarchy levels significantly improve over the classical continuous SDP bound at modest computational cost.

4. Applications in Discrete and Robust Optimization

MISDP formulations enable compact and strong modeling of diverse combinatorial problems:

Max-Cut and k-Cut: Exactly encode via $\pm 1$ matrix variables, diagonal/binary semidefinite constraints. Continuous relaxation is always weaker than the Lagrangian dual hierarchy (Meijer et al., 20 Jan 2025, Meijer et al., 2023).
Quadratic Minimum Spanning Tree (QMSTP): Structure captured with a global connectivity LMI and PSD matrix over edge variables. Tight relaxations (doubly nonnegative plus RLT cuts) dominate past LP, RLT, GL, and continuous SDP bounds (Meijer et al., 7 Oct 2024).
Quadratic Assignment, Graph Partitioning, and TSP: Matrix lifting techniques using association schemes reduce matrix dimensions, yielding compact MISDPs that are stronger than MILP formulations and much smaller than naive vector-lifting (Meijer et al., 2023).
Portfolio Optimization with Cardinality Constraints: MISDP relaxations solve many large instances to optimality, with a very high rate of rank-1 solutions and minimal (sub-1%) integrality gap (Wiegele et al., 2021).
Multistage Distributionally Robust Optimization: Stagewise MISDP subproblems, arising from decision-dependent moment-based ambiguity sets, allow distributionally robust models with risk constraints in facility location, solved via SDDiP and bounding by inner/outer approximations of PSD cones (Yu et al., 2020).

5. Bounds, Duality, and Tightness

MISDPs admit several bounding methodologies:

Continuous SDP relaxation: Allow all variables continuous; always underestimates (or overestimates in maximization) the true optimum.
Lagrangian dual: By partially dualizing constraints, always tightens the continuous relaxation. Theoretical sandwich: $z_{\text{SDP}} \leq z_{\text{LD}} \leq z_{\text{MISDP}}$ .
Hierarchical packings: Imposing integrality/PSD constraints on small submatrices (“blocks” or “packings”) yields a monotonic sequence of bounds converging to the MISDP value as block size increases (Meijer et al., 20 Jan 2025).
Convex hull DP reformulations: For MIQCQP and related classes, convex-hull representations via auxiliary binary variables yield exact formulations where the continuous convex relaxation matches the convex hull of the original feasible set (Li, 2015).

Tightness is often observed empirically; e.g., in cardinality-constrained portfolio optimization (Wiegele et al., 2021), 96% of SDP relaxations produced rank-1 solutions, i.e., global optima.

6. Computational Perspectives and Scalability

Empirical and algorithmic studies show:

MISDP relaxations can be solved by modern interior-point or splitting methods for medium and large problem sizes—up to 400-dimensional portfolios (Wiegele et al., 2021) and QMSTP with up to 50 nodes and 1,225 edges (Meijer et al., 7 Oct 2024).
Tailored splitting methods (e.g., Peaceman–Rachford for two-block DNN+cut QMSTP) scale much better than generic SDP algorithms (Meijer et al., 7 Oct 2024).
Decomposition approaches enable parallel solution of SDP subproblems and have been shown viable for large real power systems in unit commitment (Gambella et al., 2018).
Hierarchical dual bounding is especially attractive for routine bounding of large graphs; closing 30–70% of the SDP–MISDP gap with dual algorithms requiring merely small-scale local enumeration and repeated PSD projection (Meijer et al., 20 Jan 2025).
Compact modeling via matrix-lifting or association schemes is essential to limit matrix sizes and maintain tractability when problem dimensions scale (Meijer et al., 2023).
For distributionally robust and multistage settings, tightness is preserved and gap reduced to within 4% even for nontrivial (I=3, J=6, T=6, K=50) instances using MISDP-based cuts (Yu et al., 2020).

7. Future Directions and Open Problems

Cutting Plane Development: Incorporation of problem-specific cutting planes exploiting the algebraic structure of discrete PSD cones, Chvátal–Gomory procedures for conic sets, and optimization over facially reduced cones (Meijer et al., 7 Oct 2024, Meijer et al., 2023).
Generalization of Integrality: Extending theory and modeling to encompass integer valued variables beyond $\{0,1\}$ and $\{\pm 1\}$ , and richer association schemes.
Heuristic and Global Algorithms: Integration of sparse or overlapping packings into dual bounding, and advanced B&B strategies leveraging dual hierarchies and convex hull MISDPs for global certificates (Meijer et al., 20 Jan 2025, Li, 2015).
Applications in Robust and Stochastic Optimization: Extension to mixed-integer polynomial optimization, robust control, sensor network localization, and further exploitation of decision-dependent ambiguity modeling (Yu et al., 2020).
Parallel and Distributed Algorithms: Efficient distributed first-order and splitting approaches for very large-scale MISDPs, particularly leveraging problem decomposition and rapid local enumeration (Meijer et al., 20 Jan 2025, Meijer et al., 7 Oct 2024).

MISDP thus constitutes a unifying, structurally expressive, and computationally leveraged paradigm for discrete and robust optimization, combining the strengths of SDP relaxations, discrete matrix theory, and algorithmic advances in integer and conic programming.