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Probabilistic Approach to Black-Box Binary Optimization with Budget Constraints: Application to Sensor Placement

Ahmed Attia

TL;DR

The paper addresses budget-constrained binary optimization with black-box objectives by reframing the problem as a probabilistic policy optimization over conditional Bernoulli models. By conditioning the Bernoulli design on budget feasibility and optimizing the expected objective, it yields an optimal sampling policy that concentrates search within the feasible region, reducing computational waste. The work introduces and analyzes PB, CB, and GCB probability models, derives exact and stochastic gradients, proves convergence properties, and demonstrates scalability on bilinear test problems and sensor-placement/OED tasks. The approach delivers a plug-and-play tool for large-scale decision problems where objective evaluations are expensive, enabling hard constraint satisfaction without penalty-based relaxations and leveraging data-parallel sampling to tackle high-dimensional design spaces.

Abstract

We present a fully probabilistic approach for solving binary optimization problems with black-box objective functions and with budget constraints. In the probabilistic approach, the optimization variable is viewed as a random variable and is associated with a parametric probability distribution. The original optimization problem is replaced with an optimization over the expected value of the original objective, which is then optimized over the probability distribution parameters. The resulting optimal parameter (optimal policy) is used to sample the binary space to produce estimates of the optimal solution(s) of the original binary optimization problem. The probability distribution is chosen from the family of Bernoulli models because the optimization variable is binary. The optimization constraints generally restrict the feasibility region. This can be achieved by modeling the random variable with a conditional distribution given satisfiability of the constraints. Thus, in this work we develop conditional Bernoulli distributions to model the random variable conditioned by the total number of nonzero entries, that is, the budget constraint. This approach (a) is generally applicable to binary optimization problems with nonstochastic black-box objective functions and budget constraints; (b) accounts for budget constraints by employing conditional probabilities that sample only the feasible region and thus considerably reduces the computational cost compared with employing soft constraints; and (c) does not employ soft constraints and thus does not require tuning of a regularization parameter, for example to promote sparsity, which is challenging in sensor placement optimization problems. The proposed approach is verified numerically by using an idealized bilinear binary optimization problem and is validated by using a sensor placement experiment in a parameter identification setup.

Probabilistic Approach to Black-Box Binary Optimization with Budget Constraints: Application to Sensor Placement

TL;DR

The paper addresses budget-constrained binary optimization with black-box objectives by reframing the problem as a probabilistic policy optimization over conditional Bernoulli models. By conditioning the Bernoulli design on budget feasibility and optimizing the expected objective, it yields an optimal sampling policy that concentrates search within the feasible region, reducing computational waste. The work introduces and analyzes PB, CB, and GCB probability models, derives exact and stochastic gradients, proves convergence properties, and demonstrates scalability on bilinear test problems and sensor-placement/OED tasks. The approach delivers a plug-and-play tool for large-scale decision problems where objective evaluations are expensive, enabling hard constraint satisfaction without penalty-based relaxations and leveraging data-parallel sampling to tackle high-dimensional design spaces.

Abstract

We present a fully probabilistic approach for solving binary optimization problems with black-box objective functions and with budget constraints. In the probabilistic approach, the optimization variable is viewed as a random variable and is associated with a parametric probability distribution. The original optimization problem is replaced with an optimization over the expected value of the original objective, which is then optimized over the probability distribution parameters. The resulting optimal parameter (optimal policy) is used to sample the binary space to produce estimates of the optimal solution(s) of the original binary optimization problem. The probability distribution is chosen from the family of Bernoulli models because the optimization variable is binary. The optimization constraints generally restrict the feasibility region. This can be achieved by modeling the random variable with a conditional distribution given satisfiability of the constraints. Thus, in this work we develop conditional Bernoulli distributions to model the random variable conditioned by the total number of nonzero entries, that is, the budget constraint. This approach (a) is generally applicable to binary optimization problems with nonstochastic black-box objective functions and budget constraints; (b) accounts for budget constraints by employing conditional probabilities that sample only the feasible region and thus considerably reduces the computational cost compared with employing soft constraints; and (c) does not employ soft constraints and thus does not require tuning of a regularization parameter, for example to promote sparsity, which is challenging in sensor placement optimization problems. The proposed approach is verified numerically by using an idealized bilinear binary optimization problem and is validated by using a sensor placement experiment in a parameter identification setup.
Paper Structure (39 sections, 19 theorems, 126 equations, 20 figures, 3 algorithms)

This paper contains 39 sections, 19 theorems, 126 equations, 20 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background: Unconstrained Binary Optimization via Stochastic Learning
  3. Probability Models
  4. Preliminary relations
  5. Inclusion probability
  6. Efficient evaluation of the R-function
  7. Derivatives of the R-function
  8. Poisson binomial distribution
  9. The degenerate case
  10. The conditional Bernoulli distribution
  11. Moments of the CB model
  12. The degenerate case
  13. Generalized conditional Bernoulli model
  14. Sampling
  15. Sampling the PB distribution
  16. ...and 24 more sections

Key Result

Lemma 3.1

\newlabellemma:grad_log_weights0 Let $\mathbf{{\zeta}}\in\{0, 1\}^{n_{\rm s}}$ be a Bernoulli random vector with parameter $\mathbf{{\theta}}\in(0, 1)^{n_{\rm s}}$, and consider the vector of weights $\mathbf{{w}} := (w_1,\ldots,w_{n_{\rm s}})^{\mkern-1.5mu\mathsf{T}}$ with entries defined by eqn: where $\mathbf{{I}}$ is the identity matrix and $\mathsf{diag}\left(\mathbf{{w}}\right)$ is a diagon

Figures (20)

  • Figure 1: Recursive generation of $R(n, S),\, n=2,\, S:=\{1, 2, 3, n_{\rm s}=4\}$ by using the relation \ref{['eqn:R_function_and_Bernoulli_weights_recurrence_relation_2_formula']}.
  • Figure 1: Bernoulli trials weights \ref{['eqn:R_function_and_Bernoulli_weights']}. Left: weights $w$ as a function of the success probabilities $p$. Middle: values of $\frac{(1+w)^2}{w}$ for values of $\mathbf{{\theta}}\in (0, 1)$. Right: values of $\frac{w^2-1}{1+w^2}$ for values of $\mathbf{{\theta}}\in (0, 1)$.
  • Figure 1: Behavior of \ref{['alg:probabilistic_binary_optimization']} for solving \ref{['eqn:bilinear_optimization_equality']} over consecutive iterations. Left: Sample-based estimate $\widetilde{\mathbb{E}_{}{\Bigl[ \mathcal{U} \Bigr]} }$ of the stochastic objective $\Upsilon$ at each iteration of the optimization procedures. The estimate is produced by averaging $\mathcal{U}(\mathbf{{\zeta}})$ over the sample used to estimate the gradient (of size $100$) that is sampled from the distribution with parameter $\mathbf{{\theta}}$ at each iteration. Additionally, the best (largest) value of $\mathcal{U}$ among the sample is plotted. A violin plot is created for a sample of $\mathbf{{\zeta}}$ of size $100$ generated randomly from the feasible region with probability $\mathbf{{\theta}}_i=0\,\forall i=1,\ldots,n_{\rm s}=20$. Right: The value of each entry of the parameter $\mathbf{{\theta}}$ over successive iterations.
  • Figure 2: Recursive generation of $\nabla_{w}R(n, S),\, n=2,\, S:=\{1, 2, 3, n_{\rm s}=4\}$, where $R(n, S),\, n=2,\, S:=\{1, 2, 3, n_{\rm s}=4\}$ is obtained as described in \ref{['fig:R_function_recurrence_2_tabulation']}.
  • Figure 2: Results of applying \ref{['alg:probabilistic_binary_optimization']} to solve \ref{['eqn:bilinear_optimization_equality']} compared with the brute-force search of all feasible realizations of $\mathbf{{\zeta}}$. All feasible $\mathbf{{\zeta}}$ are associated with a unique integer index (on the x-axis) by using the indexing scheme in \ref{['eqn:joint_Bernoulli_expectation']}, and the corresponding value of $\mathcal{U}$ is shown on the y-axis. Brute-force results are shown as blue dots. The sample (of size $100$) generated from the policy upon termination of the optimization procedure is identified by red stars. The optimal solution (the best realization of $\mathbf{{\zeta}}$ among the sample) is identified by a red square. Additionally, the best sample point (the realization of $\mathbf{{\zeta}}$ that corresponds to the highest value of $\mathcal{U}$) that the optimizer sampled during the whole procedure is identified by a red triangle.
  • ...and 15 more figures

Theorems & Definitions (48)

  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Corollary 3.3
  • Lemma 3.4
  • Proof 3
  • Corollary 3.5
  • Lemma 3.6
  • Proof 4
  • ...and 38 more