Compact Linearization in Optimization Models

• Compact Linearization is a reformulation technique that replaces nonlinear bilinear and quadratic terms using existing structural constraints to achieve compact and robust relaxations.
• It exploits assignment and positive linear constraints to derive equivalent linear formulations with fewer inequalities than classical methods.
• The approach extends to BMI-constrained control design and composite optimization, unifying several classical reformulation frameworks in optimization.

Searching arXiv for relevant papers on compact linearization across control BMIs, binary quadratic optimization, and recent linearization formulations. Compact linearization is a structured reformulation technique for bilinear and quadratic optimization models in which existing algebraic structure is used to replace nonlinear terms by smaller linear, convex, or LMI-representable constraints. In the papers considered here, the term appears principally in two settings. In binary quadratic optimization, compact linearization replaces the standard “one product, three inequalities” construction by equations or inequalities obtained from multiplying existing linear constraints by selected binary variables, often yielding fewer constraints and, in important special cases, an LP relaxation that is provably at least as strong as the classical linearization (Mallach, 2016, Mallach, 2017, Mallach, 2018). In BMI-constrained control design, bilinear matrix mappings are decomposed as differences of positive semidefinite convex mappings, after which the concave part is linearized at each iterate, producing a sequence of convex SDP or LMI subproblems (Dinh et al., 2011).

1. Fundamental idea and canonical formulations

In binary quadratic problems, the starting point is a model with binary variables $x_i\in\{0,1\}$ and product variables $y_{ij}=x_i x_j$. The classical Glover-Woolsey, or standard, linearization introduces $y_{ij}\in[0,1]$ and imposes

$$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$

This construction is always applicable, but it adds three constraints per product and treats each product separately (Mallach, 2016, Mallach, 2018).

Compact linearization replaces that local treatment by a structure-exploiting one. For assignment constraints of the form

$\sum_{i\in A_k} x_i = 1,$

Liberti’s idea, as formalized and corrected later, is to choose sets $B_k\subseteq N$, multiply each assignment equation by $x_j$ for $j\in B_k$, and then substitute $x_i x_j$ by $y_{ij}$. This yields equations of the form

$y_{ij}=x_i x_j$0

and, after ordering product indices,

$y_{ij}=x_i x_j$1

The set $y_{ij}=x_i x_j$2 collects the product pairs actually induced by the construction. The method is compact because one multiplied assignment equation can simultaneously control many products (Mallach, 2016).

The same principle extends beyond assignment constraints. For arbitrary linear equations with positive coefficients,

$y_{ij}=x_i x_j$3

one multiplies by selected variables $y_{ij}=x_i x_j$4, obtaining

$y_{ij}=x_i x_j$5

After replacing products by $y_{ij}=x_i x_j$6, the resulting equations encode the linkage between original variables and bilinear terms more economically than the standard formulation (Mallach, 2017). A further extension also permits linear inequalities with positive coefficients and positive right-hand sides, and multiplies them either by $y_{ij}=x_i x_j$7 or by $y_{ij}=x_i x_j$8 (Mallach, 2018).

2. Assignment-constrained binary quadratic programs

The 2016 treatment considers binary quadratic programs of the form

$y_{ij}=x_i x_j$9

subject to assignment constraints

$y_{ij}\in[0,1]$0

additional linear constraints $y_{ij}\in[0,1]$1, and bilinear identities $y_{ij}\in[0,1]$2 for $y_{ij}\in[0,1]$3 (Mallach, 2016).

A central contribution is the correction of an inconsistency in Liberti’s 2007 condition. Liberti required $y_{ij}\in[0,1]$4 and $y_{ij}\in[0,1]$5 for all $y_{ij}\in[0,1]$6. The later analysis shows that while $y_{ij}\in[0,1]$7 is necessary, the condition $y_{ij}\in[0,1]$8 is not sufficient to guarantee a correct linearization in all cases, because it may enforce $y_{ij}\in[0,1]$9 without also enforcing $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$0. The revised criterion is bidirectional coverage: for each $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$1, there must exist a $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$2 such that

$$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$3

and an $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$4 such that

$$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$5

Under these two conditions, the compact equations imply

$$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$6

for each $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$7 (Mallach, 2016).

The paper also emphasizes two conceptual consequences. First, $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$8 is not necessary. Second, the true validity requirement is not set inclusion but the existence of equations in both directions for every induced product. This turns compact linearization from a heuristic recipe into a theorem-backed reformulation scheme (Mallach, 2016).

For constructing a small valid formulation, the paper introduces a mixed-integer linear program with binary variables $$y_{ij}\le x_i,\qquad y_{ij}\le x_j,\qquad y_{ij}\ge x_i+x_j-1.$$9, indicating whether $\sum_{i\in A_k} x_i = 1,$0, and continuous variables $\sum_{i\in A_k} x_i = 1,$1, indicating whether $\sum_{i\in A_k} x_i = 1,$2. The objective minimizes a weighted sum of the number of added equations and the number of additional linearization variables. When the assignment supports are pairwise disjoint,

$\sum_{i\in A_k} x_i = 1,$3

the constraint matrix is totally unimodular, so the LP relaxation already yields integer solutions. In that case, the minimally-sized linearization can be computed in polynomial time, and the associated combinatorial closure algorithm is exact with an $\sum_{i\in A_k} x_i = 1,$4 bound. When supports overlap, the same procedure remains available as a heuristic (Mallach, 2016).

3. Generalization to positive equations and inequalities

The 2017 generalization enlarges the scope from assignment constraints to arbitrary linear equations with positive coefficients,

$\sum_{i\in A_k} x_i = 1,$5

in binary quadratic problems with additional linear constraints $\sum_{i\in A_k} x_i = 1,$6. An important structural assumption is that every variable appearing in a bilinear term must appear in at least one of the linear equations (Mallach, 2017).

The construction again multiplies each equation by selected variables $\sum_{i\in A_k} x_i = 1,$7, for $\sum_{i\in A_k} x_i = 1,$8, producing

$\sum_{i\in A_k} x_i = 1,$9

and after substitution

$B_k\subseteq N$0

Here $B_k\subseteq N$1 is the set of bilinear terms induced by the multiplication process. The same two consistency conditions reappear: for each $B_k\subseteq N$2, there must be one equation with $B_k\subseteq N$3 and another with $B_k\subseteq N$4. Theorem 1 then states that for any integer solution $B_k\subseteq N$5, the standard product inequalities hold for all $B_k\subseteq N$6 (Mallach, 2017).

The strong LP-relaxation guarantee is not universal in this more general setting. For arbitrary positive coefficients and right-hand sides, the theorem guarantees the standard inequalities for integer solutions, not fractional ones. Two special cases recover the stronger statement for all $B_k\subseteq N$7. The first is the assignment case

$B_k\subseteq N$8

where the compact linearization is provably at least as strong as ordinary linearization. The second is the case

$B_k\subseteq N$9

for which the same inequalities hold for all $x_j$0, $x_j$1 (Mallach, 2017).

The 2018 paper extends the framework further to arbitrary linear equations and inequalities with positive coefficients and positive right-hand sides: $x_j$2 For equations it uses sets $x_j$3, and for inequalities two multiplier sets,

$x_j$4

Variables in $x_j$5 multiply the constraint by $x_j$6, while variables in $x_j$7 multiply it by $x_j$8. After substitution, the formulation includes equalities, upper-bounding inequalities, and complementary inequalities of the form

$x_j$9

Consistency is characterized by three conditions. Conditions 1 and 2 guarantee the zero-product cases, while Condition 3 covers the complementary case needed to force $j\in B_k$0 when both $j\in B_k$1. The paper states that, for any integer $j\in B_k$2, the linearization constraints imply $j\in B_k$3 for all $j\in B_k$4 if and only if Conditions 1–3 are satisfied (Mallach, 2018).

4. Relaxation strength, size minimization, and representative applications

A recurring question is when compact linearization is not merely valid on integer points but also at least as strong as the classical relaxation. The assignment case is the canonical answer: when every equation is

$j\in B_k$5

the induced constraints imply

$j\in B_k$6

even for fractional $j\in B_k$7, provided the bidirectional coverage conditions hold (Mallach, 2016, Mallach, 2017, Mallach, 2018).

The 2018 extension identifies further strong cases. For knapsack inequalities

$j\in B_k$8

Conditions 1–3 again imply the three classical inequalities for the relaxation. For double-selection equations

$j\in B_k$9

with products among variables in $x_i x_j$0 and $x_i x_j$1, the compact linearization also yields a relaxation at least as strong as classical linearization (Mallach, 2018).

Automatic size minimization is handled through mixed-integer models. In the assignment and positive-equation settings, variables $x_i x_j$2 indicate whether variable $x_i x_j$3 is placed in $x_i x_j$4, while variables $x_i x_j$5 indicate whether a bilinear term belongs to the induced set. The objective minimizes both the number of added equations and, secondarily, the number of additional variables. The constraints enforce inclusion of all original products and the coverage conditions needed for validity. When the supports $x_i x_j$6 are pairwise disjoint, this model reduces to a linear program with a totally unimodular matrix (Mallach, 2016, Mallach, 2017).

The framework also unifies several classical formulations from combinatorial optimization. For the quadratic assignment problem, the well-known linearization by Frieze and Yadegar for the Koopmans-Beckmann formulation is exactly a compact linearization instance; the 2018 paper also notes that Adams-Johnson formulations arise in the same manner (Mallach, 2017, Mallach, 2018). For the symmetric quadratic traveling salesman problem, degree equations

$x_i x_j$7

generate exactly the products $x_i x_j$8 when one chooses $x_i x_j$9, and after removing diagonal square terms one obtains the linearization used by Fischer and Helmberg (Mallach, 2017, Mallach, 2018). These examples are significant because they show that compact linearization is not merely a new reformulation device; it also supplies a unifying explanation for several models that had previously appeared problem by problem.

5. Convex-concave compact linearization for bilinear matrix inequalities

In BMI-constrained control design, compact linearization takes a different but related form. The 2011 paper studies optimization problems

$y_{ij}$0

where $y_{ij}$1 is convex, $y_{ij}$2 is a closed convex set, and $y_{ij}$3 are psd-convex matrix-valued mappings. The feasible set is assumed to satisfy

$y_{ij}$4

so there exists a strictly feasible starting point (Dinh et al., 2011).

The core device is to rewrite bilinear matrix terms as differences of positive semidefinite convex mappings. For the bilinear form

$y_{ij}$5

Lemma 3.1 gives explicit decompositions: $y_{ij}$6

$y_{ij}$7

and

$y_{ij}$8

Thus the bilinear mapping is written as

$y_{ij}$9

with both $y_{ij}=x_i x_j$00 and $y_{ij}=x_i x_j$01 psd-convex (Dinh et al., 2011).

At iteration $y_{ij}=x_i x_j$02, the concave part is replaced by its first-order upper bound,

$y_{ij}=x_i x_j$03

so the original constraint

$y_{ij}=x_i x_j$04

is conservatively approximated by

$y_{ij}=x_i x_j$05

This yields the convex subproblem

$y_{ij}=x_i x_j$06

The quadratic regularization term is used to stabilize the iteration and help ensure descent (Dinh et al., 2011).

A major advantage is that, in the static output feedback applications considered, many of the resulting constraints can be converted to LMIs by Schur complements. The plant is

$y_{ij}=x_i x_j$07

with static controller $y_{ij}=x_i x_j$08, giving closed-loop matrix

$y_{ij}=x_i x_j$09

Problems including stabilization, spectral abscissa, $y_{ij}=x_i x_j$10, $y_{ij}=x_i x_j$11, and mixed $y_{ij}=x_i x_j$12 synthesis become BMI-constrained optimization problems. For sparse static output feedback, the BMI constraint

$y_{ij}=x_i x_j$13

is rewritten using

$y_{ij}=x_i x_j$14

After linearizing the concave term, the constraint becomes

$y_{ij}=x_i x_j$15

which is then rewritten as an LMI via the Schur complement (Dinh et al., 2011).

The method is local. If the starting point is feasible and strict feasibility holds, every iterate remains feasible, and no line search is required. Under assumptions A1–A3, accumulation points are KKT points of the original BMI problem. The convergence theorem states that if $y_{ij}=x_i x_j$16 is strongly convex or the regularization is fixed and $y_{ij}=x_i x_j$17 has full row rank, then every accumulation point is a KKT point; if KKT points are finite, the whole sequence converges. The monotonicity relation

$y_{ij}=x_i x_j$18

shows descent (Dinh et al., 2011).

Numerically, the paper benchmarks the algorithm on COMPleib for sparse static output feedback, spectral abscissa minimization, $y_{ij}=x_i x_j$19 synthesis, $y_{ij}=x_i x_j$20 synthesis, and mixed $y_{ij}=x_i x_j$21 synthesis. The comparisons include HIFOO, PENBMI, and, in some cases, LMIRank. The reported outcomes indicate that the method often finds feasible controllers with competitive or better objective values, and in the sparse output-feedback case the paper reports a decay rate $y_{ij}=x_i x_j$22, much better than the comparison in 14.

6. Related structured linearization frameworks

Two later works show that the broader research program around compactness and linearization extends beyond product reformulation. In fully composite optimization,

$y_{ij}=x_i x_j$23

the 2023 paper proposes first-order methods that linearize only the differentiable inner map $y_{ij}=x_i x_j$24 while keeping the outer function $y_{ij}=x_i x_j$25 exact in the subproblem: $y_{ij}=x_i x_j$26 The framework generalizes Frank-Wolfe and Conditional Gradient Sliding, uses a stronger linear minimization oracle of the form

$y_{ij}=x_i x_j$27

and provides affine-invariant analysis with global convergence rates for both convex and non-convex objectives (Vladarean et al., 2023).

A different line of work studies compact linear programs generated from algorithmic descriptions. The 2025 paper introduces Hierarchical Synchronization Barriers, which decompose a program into nested Synchronization Blocks equipped with execution time intervals. The compiler then generates control-flow and memory-update constraints only on the relevant intervals, creates variable versions only for blocks that access a given variable, and uses the Union Execution Time Interval Generator to localize carry-forward constraints. On benchmark makespan and weighted minimum spanning tree instances, the paper reports up to a $y_{ij}=x_i x_j$28-fold reduction in LP size, reductions of about $y_{ij}=x_i x_j$29 and $y_{ij}=x_i x_j$30 in nonzeros and constraints on the largest tested makespan and MST instances, and substantial improvements in solver performance across cplex, gurobi, and scip (Khosravi et al., 16 Sep 2025).

These adjacent developments suggest a broader interpretation of compact linearization as a family of structure-exploiting transformations. In one branch, the structure is algebraic and comes from assignment equations, positive linear constraints, or psd-convex decompositions; in another, it is algorithmic and comes from execution regions, synchronization barriers, or composite objective structure. Across these branches, the common objective is the same: to replace an unwieldy nonlinear or overly large formulation by a smaller representation that preserves the intended optimization semantics (Vladarean et al., 2023, Khosravi et al., 16 Sep 2025).