Linear Programs with Two-Sided Constraints

• Linear programs with two-sided constraints are optimization models that impose both lower and upper bounds on decision variables, applicable across finite-dimension, Lebesgue-space, and probabilistic settings.
• The work emphasizes how geometric structures and KKT theory, even in spaces with empty interiors, can guarantee multiplier existence under strict feasibility conditions.
• Under Gaussian uncertainty, two-sided chance constraints are shown to be convex and SOCP-representable, offering precise approximations and scalable algorithmic implementations.

Linear programs with two-sided constraints are optimization models in which a decision variable, or an affine quantity derived from it, is constrained simultaneously by a lower bound and an upper bound. In finite dimensions, a canonical form is

$$\begin{aligned} &\min_{x\in\mathbb R^m} && c^\top x\ &\text{subject to} && A^\top x=b,\ & && \ell_i\le x_i\le u_i,\quad i=1,\dots,m, \end{aligned}$$

where $A\in\mathbb R^{m\times n}$, $c\in\mathbb R^m$, $b\in\mathbb R^n$, and $\ell_i<u_i$ for all $i$. In Lebesgue spaces, the analogous structure is a pointwise box $x_a\le x\le x_b$ almost everywhere together with finitely many additional linear constraints. Under Gaussian uncertainty, the same two-sided logic appears in the probabilistic condition $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$. These formulations share a common boundedness motif but lead to distinct questions about geometry, KKT theory, SOCP representability, and scalable algorithms (Gholizadeh et al., 3 Oct 2025, Wachsmuth, 2022, Lubin et al., 2015).

1. Canonical formulations

Two-sided constraints arise in at least three technically different settings. The finite-dimensional deterministic LP uses explicit coordinatewise bounds $\ell\le x\le u$. The infinite-dimensional Lebesgue-space model replaces coordinates by measurable functions and imposes $x_a(\omega)\le x(\omega)\le x_b(\omega)$ $A\in\mathbb R^{m\times n}$0-a.e. The Gaussian chance-constrained model places a lower and upper probability band on a scalar random affine form rather than on the variable itself (Gholizadeh et al., 3 Oct 2025, Wachsmuth, 2022, Lubin et al., 2015).

Setting	Formulation	Structural feature
Finite-dimensional LP	$A\in\mathbb R^{m\times n}$1 s.t. $A\in\mathbb R^{m\times n}$2	Box constraints on coordinates
Lebesgue-space program	$A\in\mathbb R^{m\times n}$3 s.t. $A\in\mathbb R^{m\times n}$4 $A\in\mathbb R^{m\times n}$5-a.e., $A\in\mathbb R^{m\times n}$6, $A\in\mathbb R^{m\times n}$7	Pointwise bounds with finitely many additional constraints
Gaussian chance model	$A\in\mathbb R^{m\times n}$8	Two-sided probabilistic bound

In the Lebesgue-space formulation, one fixes a $A\in\mathbb R^{m\times n}$9-finite measure space $c\in\mathbb R^m$0 and $c\in\mathbb R^m$1, with conjugate exponent $c\in\mathbb R^m$2. The ambient space is $c\in\mathbb R^m$3, using the norm topology for $c\in\mathbb R^m$4 and the weak-$c\in\mathbb R^m$5 topology for $c\in\mathbb R^m$6. Given measurable bounds $c\in\mathbb R^m$7, finite families $c\in\mathbb R^m$8, and scalars $c\in\mathbb R^m$9, the feasible set is $b\in\mathbb R^n$0, where

$b\in\mathbb R^n$1

and $b\in\mathbb R^n$2 denotes the finite-dimensional linear constraints (Wachsmuth, 2022).

In the Gaussian chance-constrained setting, $b\in\mathbb R^n$3 is an $b\in\mathbb R^n$4-dimensional Gaussian random vector with known covariance $b\in\mathbb R^n$5. Given $b\in\mathbb R^n$6, $b\in\mathbb R^n$7, and $b\in\mathbb R^n$8, the model is

$b\in\mathbb R^n$9

Equivalently, with $\ell_i<u_i$0 and $\ell_i<u_i$1,

$\ell_i<u_i$2

This reformulation is central to convexity and SOCP arguments (Lubin et al., 2015).

2. Geometric structure of pointwise two-sided bounds

For Lebesgue-space programs, the central geometric object is the box $\ell_i<u_i$3. A point $\ell_i<u_i$4 is called a Slater point if

$\ell_i<u_i$5

$\ell_i<u_i$6

and

$\ell_i<u_i$7

The defining feature is that $\ell_i<u_i$8 need not lie in the interior of $\ell_i<u_i$9; indeed, for $i$0, $i$1 has empty interior in $i$2 (Wachsmuth, 2022).

The normal cone to the box at $i$3 is characterized pointwise: $i$4 This yields the familiar sign pattern of lower-active, inactive, and upper-active coordinates, but now in the dual function space rather than in $i$5 (Wachsmuth, 2022).

A key structural statement is that the box set $i$6 is “$i$7-polyhedric” for all $i$8. Although $i$9, one still has

$x_a\le x\le x_b$0

The strict box bounds supplied by a Slater point are then used to prove closure of the sum $x_a\le x\le x_b$1, so that

$x_a\le x\le x_b$2

This is the geometric mechanism behind multiplier existence in the absence of genuine interior points (Wachsmuth, 2022).

3. KKT systems, multiplier existence, and necessity of Slater-type strict feasibility

If $x_a\le x\le x_b$3 is Fréchet-differentiable, and if $x_a\le x\le x_b$4 one additionally assumes $x_a\le x\le x_b$5 for all feasible $x_a\le x\le x_b$6, then a local minimizer $x_a\le x\le x_b$7 admits Lagrange multipliers provided a Slater point exists. The multiplier-existence theorem gives

$x_a\le x\le x_b$8

such that

$x_a\le x\le x_b$9

The box complementarity conditions are

$P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$0

$P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$1

$P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$2

Moreover, one may write $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$3 with nonnegative lower and upper multipliers satisfying

$P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$4

No duality gap arises, and the multipliers characterize the normal cone through

$P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$5

under the Slater hypothesis (Wachsmuth, 2022).

The proof structure is first-order: local minimality implies $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$6, and the closed-sum identity provides representation by $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$7, $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$8, and $P[a\le x^\top \xi+b\le u]\ge 1-\epsilon$9. The nontrivial point is not derivation of KKT once normal-cone additivity is available, but the closure of the sum $\ell\le x\le u$0 when $\ell\le x\le u$1 has empty interior (Wachsmuth, 2022).

The same framework also shows that Slater-type strict feasibility is, in a precise sense, necessary. Under mild hypotheses such as $\ell\le x\le u$2 and $\ell\le x\le u$3 non-atomic, if no Slater point exists then for every feasible $\ell\le x\le u$4, the sum $\ell\le x\le u$5 fails to be closed in $\ell\le x\le u$6. Consequently there exist linear functionals

$\ell\le x\le u$7

so that $\ell\le x\le u$8 is a minimizer of the linear problem $\ell\le x\le u$9 but there is no multiplier representation. This rules out the misconception that strict pointwise feasibility is merely a technical convenience in the infinite-dimensional box setting (Wachsmuth, 2022).

4. Convexity and SOCP representations for two-sided linear chance constraints

For Gaussian uncertainty, the feasible set

$x_a(\omega)\le x(\omega)\le x_b(\omega)$0

is convex when $x_a(\omega)\le x(\omega)\le x_b(\omega)$1. The argument proceeds by introducing the standard-Gaussian confidence set

$x_a(\omega)\le x(\omega)\le x_b(\omega)$2

and showing by log-concavity of the Gaussian density that $x_a(\omega)\le x(\omega)\le x_b(\omega)$3 is convex. For $x_a(\omega)\le x(\omega)\le x_b(\omega)$4,

$x_a(\omega)\le x(\omega)\le x_b(\omega)$5

and $x_a(\omega)\le x(\omega)\le x_b(\omega)$6. Hence

$x_a(\omega)\le x(\omega)\le x_b(\omega)$7

Passing to the conic hull and introducing $x_a(\omega)\le x(\omega)\le x_b(\omega)$8 gives the convex extended formulation

$x_a(\omega)\le x(\omega)\le x_b(\omega)$9

which establishes convexity (Lubin et al., 2015).

The same model admits an exact second-order-cone reformulation. Writing $A\in\mathbb R^{m\times n}$00, the exposition states

$A\in\mathbb R^{m\times n}$01

and then introduces $A\in\mathbb R^{m\times n}$02 together with

$A\in\mathbb R^{m\times n}$03

Each of (2a) and (2b) is affine in $A\in\mathbb R^{m\times n}$04, and (2c) is an SOC constraint. After substituting $A\in\mathbb R^{m\times n}$05 and $A\in\mathbb R^{m\times n}$06, this yields an SOCP-representable reformulation (Lubin et al., 2015).

Cheap outer approximations are also available. Splitting the two-sided requirement into two one-sided constraints yields

$A\in\mathbb R^{m\times n}$07

which guarantees $A\in\mathbb R^{m\times n}$08 and is described as a “2-approximation.” Adding the tangent cut to $A\in\mathbb R^{m\times n}$09 at $A\in\mathbb R^{m\times n}$10,

$A\in\mathbb R^{m\times n}$11

produces a 3-plane polyhedral outer-approximation that guarantees $A\in\mathbb R^{m\times n}$12. Together with $A\in\mathbb R^{m\times n}$13, this is an SOCP-compatible “1.25-approximation” (Lubin et al., 2015).

The framework extends beyond basic linear chance constraints. The constraint

$A\in\mathbb R^{m\times n}$14

is exactly a two-sided chance constraint and is therefore convex and SOCP-representable. More generally, whenever $A\in\mathbb R^{m\times n}$15 is univariate convex with a unique minimizer and $A\in\mathbb R^{m\times n}$16 is convex, the condition

$A\in\mathbb R^{m\times n}$17

is convex in $A\in\mathbb R^{m\times n}$18. In AC-OPF, quadratic chance constraints such as

$A\in\mathbb R^{m\times n}$19

are generally nonconvex, but may be approximated conservatively by two absolute-value constraints and a circle constraint via the union bound (Lubin et al., 2015).

5. Interior-point methods and two-party communication protocols

In finite dimensions, the two-sided box $A\in\mathbb R^{m\times n}$20 is compatible with a path-following IPM. For each coordinate,

$A\in\mathbb R^{m\times n}$21

is a two-sided log-barrier on the open interval $A\in\mathbb R^{m\times n}$22, and the exposition describes it as a “highly 1-self-concordant” barrier. Its derivatives are

$A\in\mathbb R^{m\times n}$23

With $A\in\mathbb R^{m\times n}$24, the central-path conditions are

$A\in\mathbb R^{m\times n}$25

for some $A\in\mathbb R^{m\times n}$26, and the Newton direction is obtained from the weighted normal equations

$A\in\mathbb R^{m\times n}$27

Then

$A\in\mathbb R^{m\times n}$28

followed by the update

$A\in\mathbb R^{m\times n}$29

with a short-step size $A\in\mathbb R^{m\times n}$30 (Gholizadeh et al., 3 Oct 2025).

The two-party communication model partitions the rows of $A\in\mathbb R^{m\times n}$31. Alice holds $A\in\mathbb R^{m\times n}$32 together with the corresponding entries of $A\in\mathbb R^{m\times n}$33, Bob holds $A\in\mathbb R^{m\times n}$34 and the matching parts of $A\in\mathbb R^{m\times n}$35, both know $A\in\mathbb R^{m\times n}$36, local computation is unlimited, shared randomness is allowed, and the cost measure is total bits exchanged. The protocol adapts an IPM-based sequential algorithm for LPs with two-sided constraints to this model using techniques from distributed convex optimization (Gholizadeh et al., 3 Oct 2025).

To avoid sending the full matrix $A\in\mathbb R^{m\times n}$37, the protocol uses a sampling-based spectral approximation

$A\in\mathbb R^{m\times n}$38

of dimension $A\in\mathbb R^{m\times n}$39. Each party locally forms its part of

$A\in\mathbb R^{m\times n}$40

sends these length-$A\in\mathbb R^{m\times n}$41 vectors to a coordinator, receives

$A\in\mathbb R^{m\times n}$42

and reconstructs $A\in\mathbb R^{m\times n}$43 and $A\in\mathbb R^{m\times n}$44 locally. The method also recomputes $A\in\mathbb R^{m\times n}$45-Lewis weights

$A\in\mathbb R^{m\times n}$46

which improve the condition number in the normal equations; the two-party subroutine computes these weights in $A\in\mathbb R^{m\times n}$47 bits whenever per-row sparsity is at most $A\in\mathbb R^{m\times n}$48 (Gholizadeh et al., 3 Oct 2025).

The communication analysis gives $A\in\mathbb R^{m\times n}$49 bits per Newton step, with the main components being Lewis-weight recomputation, maintenance of the spectral approximation, and exchange of the vectors $A\in\mathbb R^{m\times n}$50. Since the IPM takes

$A\in\mathbb R^{m\times n}$51

iterations to reach $A\in\mathbb R^{m\times n}$52-accuracy, the total cost is

$A\in\mathbb R^{m\times n}$53

The abstract states that this extends the scope from one-sided to two-sided constraints while matching the sparse-matrix complexity up to logarithmic factors, and that minimum cost flow can be treated as a special case of solving linear programs with two-sided constraints (Gholizadeh et al., 3 Oct 2025).

6. Relation to classical box-constrained LP and recurring misconceptions

In $A\in\mathbb R^{m\times n}$54, the box $A\in\mathbb R^{m\times n}$55 has nonempty interior if and only if $A\in\mathbb R^{m\times n}$56 componentwise, and a Slater point is simply an interior point. This recovers the usual KKT system. In $A\in\mathbb R^{m\times n}$57 for $A\in\mathbb R^{m\times n}$58, by contrast, the interior of $A\in\mathbb R^{m\times n}$59 is empty, so classical interior-point arguments break down. Nevertheless, strict pointwise inequalities almost everywhere still suffice to close the normal-cone sum and recover full KKT theory. Complementarity slackness therefore looks formally the same in finite and infinite dimensions, but in $A\in\mathbb R^{m\times n}$60 the multiplier $A\in\mathbb R^{m\times n}$61 must be interpreted as a function in the dual space (Wachsmuth, 2022).

A second recurring misconception is that a two-sided chance constraint is just the conjunction of two one-sided chance constraints. The Gaussian theory distinguishes the exact model from such decompositions: the split formulation yields only a “2-approximation,” while the addition of a tangent cut yields a “1.25-approximation.” The exact convex representation instead relies on the geometry of the standard-Gaussian confidence set $A\in\mathbb R^{m\times n}$62, on Gaussian log-concavity, and on the condition $A\in\mathbb R^{m\times n}$63 (Lubin et al., 2015).

A third misconception is that extending one-sided LP protocols to two-sided constraints is only a notational change. The two-party IPM requires replacing the nonnegative barrier $A\in\mathbb R^{m\times n}$64 by the two-sided barrier $A\in\mathbb R^{m\times n}$65, recomputing $A\in\mathbb R^{m\times n}$66 and $A\in\mathbb R^{m\times n}$67, and adjusting the Lewis-weight definition to depend on $A\in\mathbb R^{m\times n}$68. The data indicate that, despite these conceptual changes, the overall communication remains $A\in\mathbb R^{m\times n}$69 (Gholizadeh et al., 3 Oct 2025).

Taken together, these results suggest that “linear programs with two-sided constraints” is not a single theory but a cluster of closely related theories. In finite-dimensional deterministic optimization, the main issues are barrier design and linear-system solution. In Lebesgue spaces, the decisive issue is the geometry of box constraints without interior points. Under Gaussian uncertainty, the decisive issue is convexity of a probability region and its conic representability. The common boundedness pattern is therefore algebraically simple, but its analytical consequences depend strongly on the ambient space and on whether the bounds are deterministic or probabilistic.