Global Optimization Techniques for Multiplicative Programming Problems

Multiplicative programming problems are characterised by objective functions or constraints that involve products of decision variables, giving rise to nonconvex landscapes with multiple local optima. Global optimisation techniques seek the true optimum by systematically exploring and pruning the search space. Central to these methods is the branch-and-bound framework, which partitions the domain into subregions and employs bounding procedures to eliminate regions that cannot contain the global solution. Early strategies transformed bilinear or fractional terms via convex envelopes and affine approximations, yielding linear relaxations that provide rigorous lower bounds. Outer-space search algorithms extended this idea by conducting the partitioning in an expanded auxiliary-variable space, improving bound tightness and convergence speed. Image-space approaches apply logarithmic or other nonlinear transformations to the objective, enabling direct operating on the outcome space and more aggressive region reduction. Hybrid schemes combine convex relaxation with super-rectangular or outcome-space reduction, auxiliary-variable introduction and adaptive partitioning criteria. Collectively, these advances have broadened the class of tractable problems, enhanced computational efficiency and opened applications across engineering design, resource allocation and network optimisation.

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In a comprehensive computational assessment, contemporary studies of branch-and-bound strategies for linear multiplicative programmes have refined underestimation functions and partitioning rules. By tailoring bounding functions to problem structure, significant reductions in node explorations and solution times have been achieved, extending applicability to bilevel formulations and large-scale network design. Parallel work on image-space branch–reduction–bound algorithms employs logarithmic reformulation of the objective to derive tighter parametric linear relaxations, coupled with novel pruning rules that exploit curvature in the transformed space. Benchmark comparisons demonstrate superior convergence rates and robustness in higher dimensions relative to classical bilinear methods. Complementing these, output-space branch-and-bound reduction algorithms integrate super-rectangular contraction techniques with direct relaxation bounding, enhancing lower-bound quality and ensuring global convergence. Empirical evaluations against standard test sets confirm the scalability and practical performance gains of these methods.

Global Optimization Techniques for Multiplicative Programming Problems publication trend

The graph below shows the total number of articles in global optimization techniques for multiplicative programming problems across all publications each year (not limited to Nature Index journals).

Technical terms

Multiplicative programming problem: A nonconvex optimisation problem in which the objective or constraints involve products of decision variables.

Branch-and-bound algorithm: An iterative scheme that partitions the search domain into subregions and uses bounds to prune those that cannot contain the global optimum.

Linear relaxation: An approximation technique that replaces nonlinear or nonconvex terms with linear envelopes to compute rigorous lower bounds.

Convex relaxation: A method that approximates a nonconvex problem by a convex one, facilitating bound computation and guaranteeing convergence under certain conditions.

Image space reduction: A technique that operates in a transformed outcome space, using problem structure to eliminate regions that cannot contain the global solution.

References

1. Global Optimization for Generalized Linear Multiplicative Programming Using Convex Relaxation. Mathematical Problems in Engineering (2018).
2. An Efficient Outer Space Algorithm for Generalized Linear Multiplicative Programming Problem. IEEE Access (2020).
3. Image space branch-reduction-bound algorithm for globally minimizing a class of multiplicative problems. RAIRO - Operations Research (2022).
4. Output-Space Branch-and-Bound Reduction Algorithm for a Class of Linear Multiplicative Programs. Mathematics (2020).
5. Outcome space range reduction method for global optimization of sum of affine ratios problem. Open Mathematics (2016).
6. Solving linear multiplicative programs via branch-and-bound: a computational experience. Computational Management Science (2023).