Mixed-Integer Quadratic Programming

• MIQP is an optimization paradigm that minimizes a quadratic objective over a mixed space of integer and continuous variables, encapsulating NP-hard problems like maximum cut.
• It combines the combinatorial challenges of integer programming with the nonconvex traits of quadratic programming, employing geometric and algebraic techniques to secure polynomial-size certificates.
• The approach justifies certificate-based algorithms such as branch-and-bound while highlighting limits where multiple quadratic constraints may require exponential solution representations.

Mixed-Integer Quadratic Programming (MIQP) comprises optimization problems in which a quadratic objective is minimized or maximized over a polyhedral region where a subset of variables must assume integer values. MIQP is foundational in operations research, engineering, and control, uniting mathematical challenges from both integer programming (combinatorial complexity) and quadratic programming (convex and nonconvex structures). MIQP generalizes integer linear programming and quadratic programming, encompassing NP-hard problems such as maximum cut and integer least squares. The central theoretical result is that the (single-quadratic-inequality) feasibility version is in NP, enabling certificate-based algorithmic approaches and establishing NP-completeness (Pia et al., 2014). The paper and resolution of MIQP engage geometric, algebraic, and algorithmic techniques, with the additional complexity of nonconvexity and integrality introducing practical and mathematical obstacles.

1. Problem Structure and Formal Definition

A general MIQP takes the form: $$\min_{x \in \mathbb{Z}^p \times \mathbb{R}^{n-p}} \left\{ x^T H x + c^T x + d : \; Ax \leq b \right\}$$ where $H \in \mathbb{Q}^{n \times n}$ is symmetric (possibly indefinite), $c \in \mathbb{Q}^n$, $d \in \mathbb{Q}$, $A \in \mathbb{Q}^{m \times n}$, and $b \in \mathbb{Q}^m$. The variable vector $x$ is partitioned into $p$ integer and $n-p$ continuous variables.

A canonical feasibility version is to decide whether the set

$$F(H,c,d,A,b) = \{ x \in \mathbb{Z}^p \times \mathbb{R}^{n-p} : Ax \leq b,\, x^T H x + c^T x + d \leq 0 \}$$

is nonempty.

The quadratic objective introduces nonconvexity when $H$ is not positive semidefinite, while the mixed-integer structure encodes combinatorial complexity. The feasible region is a mixed-integer slice of a (possibly unbounded) polyhedron, intersected with a quadratic level set.

2. Complexity and Certificates

A principal result is that the decision version of MIQP with a single quadratic inequality is in NP (Pia et al., 2014). More precisely, for a given instance described by $(H, c, d, A, b)$, if the feasible set $F(H, c, d, A, b)$ is nonempty, then there exists a feasible point $x^0$ whose binary encoding has length bounded by a polynomial in the size of the input data (the “complexity” $\varphi$ of the instance). The main theorem asserts:

• If $F(H, c, d, A, b)$ is nonempty, there exists $x^0 \in F(H, c, d, A, b)$ such that the complexity of $x^0$ is upper bounded by $f(\varphi)$ for a polynomial $f$.

Consequences:

• Certifying feasibility for MIQP can be performed by exhibiting a polynomial-size solution, verifiable in polynomial time.
• Since MIQP feasibility is known to be NP-hard (e.g., due to the ability to encode maximum cut, integer least squares, and other classic NP-hard problems), it follows immediately that MIQP is NP-complete.

The result generalizes prior work for quadratic programming (continuous case) and integer linear programming:

• Quadratic programming is in NP (Vavasis 1990);
• Integer linear programming is in NP (Borosh–Treybig, Papadimitriou, and others).

3. Proof Principles and Geometric Decomposition

The proof that MIQP is in NP leverages geometric, polyhedral, and algebraic techniques, focusing on two primary cases depending on the structure of the unbounded directions in the underlying polyhedron $\mathcal{C} = \{ x : Ax \leq b \}$.

a. Use of the Recession Cone

The proof assumes $\mathcal{C}$ is pointed. The recession cone $\operatorname{rec}(\mathcal{C})$ describes directions along which $Ax \leq b$ remains satisfied indefinitely. The quadratic form is examined over $\operatorname{rec}(\mathcal{C})$ intersected with a “normalizing hyperplane” $\mathcal{H} = \{ x : f^T x = 1 \}$ (Lemma 1). The hyperplane is constructed so that $f$'s bit-size is polynomially bounded.

Case 1: Strictly Negative Quadratic Growth Along a Recession Direction

If $$\min\{r^T H r : r \in \operatorname{rec}(\mathcal{C}) \cap \mathcal{H}\} < 0$$, then moving far enough in this direction decreases the quadratic value. Explicitly, for some recession ray $\tilde{r} \in \mathbb{Z}^p \times \mathbb{R}^{n-p}$ and a feasible point $\tilde{x}$, one obtains feasible points $\tilde{x} + \lambda \tilde{r}$ for $\lambda$ large. The quadratic evaluates as: $$Q(\tilde{x} + \lambda \tilde{r}) = \lambda^2 \tilde{r}^T H \tilde{r} + \lambda (2\tilde{x}^T H \tilde{r} + c^T \tilde{r}) + (c^T \tilde{x} + d)$$ For $r^T H r < 0$, a suitable $\lambda$ (computable in polynomial time) makes $Q(\cdot) < 0$. Bit-sizes are polynomially controlled.

Case 2: Nonnegative Quadratic Growth Along All Recession Directions

If $\min\{r^T H r\} \geq 0$, unbounded directions cannot be used to decrease the quadratic value sufficiently. In this case, Proposition 2 decomposes the set $$\mathcal{C} \cap (\mathbb{Z}^p \times \mathbb{R}^{n-p})$$ into a finite union of sets of the form “polytope plus integer cone”, each of polynomial description complexity: $$\mathcal{C} \cap (\mathbb{Z}^p \times \mathbb{R}^{n-p}) = \bigcup_{i, K} (P_i + \operatorname{int.cone}(R_K))$$ Within each component, the quadratic function behaves either linearly or remains bounded. When linear, if the coefficient is negative, moving along the corresponding integer directions suffices to achieve $Q(x) \leq 0$; otherwise the feasible set is bounded in that piece, so a candidate with polynomial size must exist (by Theorem QPisNP for the continuous QP case).

In all scenarios, the solution “certificate” lies within a bounded region with polynomially many bits.

4. Algorithmic and Complexity Implications

The existence of a polynomial-size certificate assures important algorithmic consequences:

• For branch-and-bound and certificate-based exact algorithms, one is guaranteed, a priori, that a succinct feasible solution exists for any yes-instance.
• For practical computation, this provides a theoretical underpinning for seeking compact witnesses (solutions), analogous to integer linear programming.
• The result is sharply bounded in terms of generalization: if the MIQP formulation permits multiple quadratic inequalities, the “small solution” property can fail (counterexamples arise in Pell’s equations or systems with many quadratic inequalities, where solution bit-size may be exponential).

5. Connections to Other Paradigms and Limitations

• MIQP strictly generalizes both integer quadratic programming (IQP) and continuous quadratic programming (QP).
• IQP, the all-integer special case, and MIQP, in turn, can encode NP-hard problems such as maximum cut.
• The result cannot be extended to the case of multiple quadratic inequalities: known negative results show that exponential-size certificates may be needed in such cases.

6. Principal Formulas and Representation

Central mathematical representations employed in the result include:

• Feasibility set:

$$F(H,c,d,A,b) = \{ x \in \mathbb{Z}^p \times \mathbb{R}^{n-p} : Ax \leq b,\, x^T H x + c^T x + d \leq 0 \}$$

• Recession cone minimization for normalization:

$$\min \{ r^T H r : r \in \operatorname{rec}(\mathcal{C}) \cap \mathcal{H} \}$$

• Quadratic value along rays:

$$Q(\tilde{x} + \lambda \tilde{r}) = \lambda^2 \tilde{r}^T H \tilde{r} + \lambda (2\tilde{x}^T H \tilde{r} + c^T \tilde{r}) + (c^T \tilde{x} + d)$$

An explicit decomposition into finitely many (polytope + integer cone) regions:

$$\mathcal{C} \cap (\mathbb{Z}^p \times \mathbb{R}^{n-p}) = \bigcup_{i,K} (P_i + \operatorname{int.cone}(R_K))$$

7. Practical Applications and Boundaries

From an applications perspective, the recognition that MIQP is in NP establishes the theoretical feasibility of algorithms predicated on certificate verification. Certificate-based algorithms (e.g., branch-and-bound, cutting planes) are justified because any feasible instance must, by construction, admit a small (polynomially-encoded) solution.

Limitation arises in general quadratic systems: with more than one quadratic constraint, the property no longer holds universally. For instance, in systems motivated by Pell’s equation or those with many simultaneous quadratic constraints, feasible solutions may necessarily require an exponential number of bits in their encoding.

8. Summary and Significance

The core results on MIQP complexity unify two major threads in mathematical optimization—quadratic programming and integer linear programming—demonstrating that any “yes”-instance of MIQP with a single quadratic inequality can be certified by a solution vector whose representation length is polynomial in the input size. This places MIQP squarely in NP-complete territory, underlining both the tractability (from the standpoint of certificate existence and verifiability) and the intractability (NP-hardness) that mark the field. The boundary of this result—single vs. multiple quadratic inequalities—is sharp and aligns with known negative results. These findings serve as a rigorous foundation for the design and justification of exact algorithms in MIQP, guiding both theoretical research and practical algorithm development (Pia et al., 2014).