ShaPeak: Sharp-Peak Functions for UBIP

• ShaPeak is an algorithmic framework for unconstrained binary integer programming that reformulates binary constraints using sharp-peak functions for an exact penalty model.
• It introduces a proximal stationarity concept and employs an inexact ADMM approach to guarantee global convergence with a linear rate under mild assumptions.
• The framework demonstrates practical efficiency in applications like recovery, MIMO detection, and QUBO through robust parameterization and closed-form proximal mappings.

ShaPeak is an algorithmic framework for unconstrained binary integer programming (UBIP) derived from a class of penalty functions called sharp-peak functions (SPFs). In the formulation studied in "Sharp-Peak Functions for Exactly Penalizing Binary Integer Programming" (Zhou et al., 31 Aug 2025), UBIP is rewritten by replacing the discrete constraint $x\in\{0,1\}^n$ with equality constraints induced by an SPF, and the resulting constrained problem is then handled through an exact penalty model. ShaPeak itself is an inexact alternating direction methods of multipliers procedure designed for that penalty model; its analysis introduces a proximal stationarity notion, proves global convergence to a P-stationary point under a mild local Lipschitz assumption on $\nabla f$, and establishes a linear rate and finite termination bound under stronger penalty initialization conditions.

1. Problem setting and sharp-peak reformulation

The underlying optimization problem is the standard UBIP

$$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$

where $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$ is continuously differentiable. The paper uses the box notation $B=\{x\in\mathbb{R}: 0\le x\le 1\}$ and $\mathcal{X}=B^n$, together with the indicator $\delta_\Omega(x)=0$ if $x\in\Omega$ and $+\infty$ otherwise. For the box constraint, the normal cone $N_{\mathcal X}(x)$ is described coordinate-wise by

$\nabla f$0

A function $\nabla f$1 is an SPF if it satisfies three conditions. First, for any $\nabla f$2, $\nabla f$3 and $\nabla f$4 if and only if $\nabla f$5. Second, $\nabla f$6 is lower semicontinuous on $\nabla f$7, and there exists $\nabla f$8 such that

$\nabla f$9

Third, the endpoint sum rule holds: $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$0 These conditions yield the “sharp peak” behavior at $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$1 and $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$2, with a uniform lower bound on subgradient magnitude.

The SPF induces the equality-constrained reformulation

$$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$3

which is equivalent to UBIP because $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$4 together with $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$5 is equivalent to $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$6. This converts discrete feasibility into a system of scalar equalities over the box $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$7 (Zhou et al., 31 Aug 2025).

The paper gives two main SPF families. One is the smooth “absolute-value peak” family

$$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$8

with $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$9, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$0, and $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$1. Examples listed are $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$2, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$3, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$4, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$5, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$6, $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$7, and $f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$8. The other consists of piecewise polynomial families

$f:\mathbb{R}^n\to\mathbb{R}\cup\{\infty\}$9

and

$B=\{x\in\mathbb{R}: 0\le x\le 1\}$0

A recurrent point of clarification is that not every function vanishing at $B=\{x\in\mathbb{R}: 0\le x\le 1\}$1 is an SPF. The paper explicitly notes that $B=\{x\in\mathbb{R}: 0\le x\le 1\}$2 with $B=\{x\in\mathbb{R}: 0\le x\le 1\}$3 and $B=\{x\in\mathbb{R}: 0\le x\le 1\}$4 do not qualify because condition c2 fails at $B=\{x\in\mathbb{R}: 0\le x\le 1\}$5.

2. Exact penalty model and threshold phenomenon

Rather than solving the SPF-constrained problem directly, the framework studies the exact penalty model

$B=\{x\in\mathbb{R}: 0\le x\le 1\}$6

and introduces the threshold

$B=\{x\in\mathbb{R}: 0\le x\le 1\}$7

where $B=\{x\in\mathbb{R}: 0\le x\le 1\}$8 is the SPF subgradient lower bound from c2. Since $B=\{x\in\mathbb{R}: 0\le x\le 1\}$9 is continuous on the bounded box $\mathcal{X}=B^n$0, $\mathcal{X}=B^n$1.

The central exact penalty theorem states that if $\mathcal{X}=B^n$2, then a point is a global minimizer of SPCO if and only if it is a global minimizer of EPM. Because SPCO is equivalent to UBIP, the global minimizers of UBIP and the penalty model coincide once $\mathcal{X}=B^n$3 exceeds this threshold (Zhou et al., 31 Aug 2025).

The paper emphasizes that this threshold is stronger than classical exact penalty constants in a specific sense: $\mathcal{X}=B^n$4 depends only on the SPF parameter $\mathcal{X}=B^n$5 and on $\mathcal{X}=B^n$6 over $\mathcal{X}=B^n$7, and is independent of the UBIP solution set. This removes the usual dependence on an unknown optimal set from the penalty parameter characterization.

The penalty viewpoint is also structurally important for the algorithm. The term $\mathcal{X}=B^n$8 is separable across coordinates, while the SPF construction makes the penalty zero exactly at binary points and strictly positive elsewhere in the box. This provides a continuous objective landscape over $\mathcal{X}=B^n$9 while preserving exact correspondence with the discrete problem when $\delta_\Omega(x)=0$0.

3. Optimality framework: KKT points and P-stationarity

For the penalty model, a point $\delta_\Omega(x)=0$1 is a Karush-Kuhn-Tucker point if

$\delta_\Omega(x)=0$2

Equivalently, coordinate-wise,

$\delta_\Omega(x)=0$3

The paper proves three basic facts: any KKT point is binary, any local minimizer is a KKT point, and if $\delta_\Omega(x)=0$4 and $\delta_\Omega(x)=0$5 are locally convex around a KKT point, then that KKT point is a local minimizer.

The analysis then introduces P-stationarity. A point $\delta_\Omega(x)=0$6 is P-stationary with stepsize $\delta_\Omega(x)=0$7 if

$\delta_\Omega(x)=0$8

where

$\delta_\Omega(x)=0$9

This notion is tailored to the proximal structure of the SPF penalty.

The relations between these optimality concepts are central to ShaPeak. Any P-stationary point is a KKT point. Conversely, any KKT point is P-stationary if $x\in\Omega$0 is locally convex around that point. Any local minimizer is P-stationary if either $x\in\Omega$1 and $x\in\Omega$2 are locally convex around the point or $x\in\Omega$3 is $x\in\Omega$4-strongly smooth on $x\in\Omega$5. In addition, if $x\in\Omega$6 is $x\in\Omega$7-strongly convex on $x\in\Omega$8 and $x\in\Omega$9, then any P-stationary point is a global minimizer of EPM (Zhou et al., 31 Aug 2025).

These statements give P-stationarity a specific algorithmic role. It is weaker than an a priori global optimality statement, but stronger than a generic fixed-point heuristic: P-stationary points are binary through the KKT implication, and under curvature assumptions on $+\infty$0 they coincide with globally optimal solutions of the penalty model.

4. ShaPeak algorithm and computational structure

ShaPeak is derived by splitting the penalty model with an auxiliary variable $+\infty$1 and multiplier $+\infty$2: $+\infty$3 The augmented Lagrangian with penalty $+\infty$4 is

$+\infty$5

The resulting procedure is an inexact ADMM scheme.

Given $+\infty$6, the $+\infty$7-update is

$+\infty$8

Because $+\infty$9, this proximal mapping splits coordinate-wise. For the piecewise families $N_{\mathcal X}(x)$0 and $N_{\mathcal X}(x)$1, the paper shows that the box-constrained proximal operators admit closed forms, and for $N_{\mathcal X}(x)$2 these become explicit threshold and linear or quadratic formulas. The paper further states that for sufficiently large proximal steps $N_{\mathcal X}(x)$3, the proximal map often returns binary values.

The $N_{\mathcal X}(x)$4-update is linearized around $N_{\mathcal X}(x)$5 and yields the closed form

$N_{\mathcal X}(x)$6

where $N_{\mathcal X}(x)$7 is a preconditioner, such as an approximation to the Hessian or $N_{\mathcal X}(x)$8 in least-squares, and the analysis assumes $N_{\mathcal X}(x)$9 for a given $\nabla f$00. The dual step is the standard ADMM multiplier update

$\nabla f$01

A distinctive component is the adaptive penalty schedule

$\nabla f$02

with parameters $\nabla f$03, $\nabla f$04, $\nabla f$05, and integer $\nabla f$06. If $\nabla f$07, then $\nabla f$08 is already binary and no increase is required.

The practical initialization recommended is $\nabla f$09 and $\nabla f$10 with $\nabla f$11. Termination uses

$\nabla f$12

The returned $\nabla f$13 is then a P-stationary point under the proximal characterization (Zhou et al., 31 Aug 2025).

5. Convergence theory, Lyapunov descent, and rate results

The convergence analysis begins from the bounded sublevel region

$\nabla f$14

which is bounded because $\nabla f$15 is continuous and $\nabla f$16 is bounded. The scalar

$\nabla f$17

satisfies $\nabla f$18 in $\nabla f$19.

The standing smoothness requirement is a mild local one: $\nabla f$20 is locally Lipschitz continuous on a slightly enlarged bounded box $\nabla f$21. On any compact subset of $\nabla f$22, this implies Lipschitz continuity with constant $\nabla f$23. The algorithmic penalty parameter is then required to satisfy

$\nabla f$24

Under this choice, the paper defines the auxiliary Lyapunov quantity

$\nabla f$25

and proves the descent inequality

$\nabla f$26

The global convergence theorem states that $\nabla f$27 and $\nabla f$28 converge to the same limit; $\nabla f$29 is monotone nondecreasing and bounded above, hence converges to $\nabla f$30; and the successive differences satisfy

$\nabla f$31

Moreover, any accumulation point $\nabla f$32 of $\nabla f$33 is a P-stationary point of $\nabla f$34 with proximal stepsize $\nabla f$35: $\nabla f$36 If $\nabla f$37, then the full sequences $\nabla f$38 and $\nabla f$39 converge to a binary P-stationary point $\nabla f$40 (Zhou et al., 31 Aug 2025).

A stronger theorem gives a linear rate and a finite iteration bound. If

$\nabla f$41

then there exists $\nabla f$42 such that for all $\nabla f$43,

$\nabla f$44

with

$\nabla f$45

Consequently, the stopping criteria are met after at most

$\nabla f$46

iterations.

6. Empirical behavior, parameterization, and limitations

The numerical study evaluates ShaPeak on four classes of problems. For recovery problems with binary variables, the objective is $\nabla f$47 with $\nabla f$48 and dimensions up to $\nabla f$49, using recovery accuracy $\nabla f$50. For classical MIMO detection, both i.i.d. and correlated channels with QPSK are tested up to $\nabla f$51, with BER as the metric. For one-bit MIMO detection, the objective is a non-quadratic likelihood with probit link, tested up to $\nabla f$52, again with BER. For QUBO, dense and sparse instances up to $\nabla f$53 are studied using relative optimality gap

$\nabla f$54

and runtime. Baselines are NPGM, MEPM, L2ADMM, EMADM, GUROBI, SDPNAL+, HOTML, and ZF/MMSE decision-feedback.

An SPF ablation compares nine variants. The piecewise-quadratic choices with $\nabla f$55, $\nabla f$56, $\nabla f$57,

$\nabla f$58

and

$\nabla f$59

are reported to have consistently delivered the best accuracy and were adopted in ShaPeakg and ShaPeakh.

The reported performance highlights are problem-dependent but consistent in direction. In recovery experiments, ShaPeakg and ShaPeakh maintained $\nabla f$60 recovery accuracy across wide ranges, were robust to noise with $\nabla f$61 up to $\nabla f$62, and often outperformed GUROBI and EMADM; for very large $\nabla f$63, MEPM and L2ADMM sometimes failed or produced poor accuracy, while ShaPeak sustained accuracy with significantly lower runtime. In classical MIMO detection, across SNR and both channel models, ShaPeak achieved the lowest BER among algorithmic baselines, with runtime typically one to two orders of magnitude faster than L2ADMM, MEPM, and HOTML; the paper gives as one example i.i.d. SNR $\nabla f$64 at $\nabla f$65, where ShaPeakg and ShaPeakh required about $\nabla f$66 seconds versus more than $\nabla f$67 seconds for the others. In one-bit MIMO detection, for $\nabla f$68 and moderate-to-high SNR $\nabla f$69, ShaPeak attained the best BER, and at SNR $\nabla f$70 its runtime was about $\nabla f$71 seconds versus $\nabla f$72–$\nabla f$73 seconds for L2ADMM and MEPM. In QUBO, GUROBI often had the lowest gap for $\nabla f$74 but under $\nabla f$75-second runtime caps, whereas ShaPeak achieved near-best gaps with runtimes such as $\nabla f$76–$\nabla f$77 seconds; for $\nabla f$78 and $\nabla f$79, ShaPeakg and ShaPeakh attained the lowest objectives among non-commercial solvers with $\nabla f$80–$\nabla f$81 seconds, compared with $\nabla f$82–$\nabla f$83 seconds for MEPM and L2ADMM (Zhou et al., 31 Aug 2025).

The parameterization guidance is explicit. The piecewise-quadratic $\nabla f$84 with $\nabla f$85, $\nabla f$86, $\nabla f$87 are recommended because they are robust and yield closed-form proximal mappings. For linear convergence and finite termination guarantees, the paper recommends $\nabla f$88; otherwise one may rely on the adaptive update. The ADMM penalty should satisfy $\nabla f$89. The preconditioner $\nabla f$90 should obey $\nabla f$91, with examples including $\nabla f$92 in least-squares or diagonalized curvature in one-bit MIMO. The adaptive schedule uses $\nabla f$93, $\nabla f$94, moderate $\nabla f$95 such as $\nabla f$96, and small $\nabla f$97, while initialization uses $\nabla f$98 and $\nabla f$99.

The stated limitations are also precise. Without strong convexity of $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$00, convergence is to P-stationary or KKT points of the penalty model; global optimality for UBIP follows from the exact penalty theorem once $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$01. The SPF sum rule c3 is required at the endpoints. The framework is presented as extendable to constrained binary integer programming by including other constraints in the ADMM splitting, and to general integer programming by designing SPFs with peaks at a discrete alphabet such as $$\min_{x\in\mathbb{R}^n}~ f(x)\quad\text{s.t.}\quad x\in\{0,1\}^n, \tag{UBIP}$$02. A plausible implication is that the core idea is not restricted to quadratic UBIP, provided the objective remains smooth enough to satisfy the local Lipschitz requirement used in the convergence proof.