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New Outer Approximation Algorithms for Nonsmooth Convex MINLP Problems

Zhou Wei, He-Yi Liu, Bo Zeng

TL;DR

The paper addresses solving nonsmooth convex MINLPs by introducing an outer-approximation framework that reformulates the problem as an equivalent MILP master $MP$ and solves a sequence of MILP relaxations with finite termination. It leverages KKT-based subgradients and a new parameter $\rho_j$ derived from nonlinear constraints to generate stronger cuts, ensuring equivalence between the MILP master and the original problem $(P)$. The method yields tighter relaxations than classical OA and demonstrates computational efficiency gains by exploiting problem structure rather than relying solely on first-order Taylor expansions. The resulting approach has practical impact for a broad class of MINLPs, offering an effective alternative to existing OA methods and enhancing solution quality and speed in convex, nonsmooth settings.

Abstract

This paper presents a novel outer approximation algorithm for nonsmooth mixed-integer nonlinear programming (MINLP) problems. The method proceeds by fixing the integer variables and solving the resulting nonlinear convex subproblem. When the subproblem is feasible, valid linear cuts are derived by computing suitable subgradients of the objective and constraint functions at the optimal solution, utilizing KKT optimality conditions. A new parameter, defined through the nonlinear constraint functions, is introduced to facilitate the generation of these cuts. For infeasible subproblems, a feasibility problem is solved, and valid linear cuts are generated via KKT-based subgradients to exclude the infeasible integer assignment. By integrating both types of cuts, a mixed-integer linear programming (MILP) master problem is formulated and proven equivalent to the original MINLP. This equivalence underpins a new outer approximation algorithm, which is guaranteed to terminate after a finite number of iterations. Numerical experiments on smooth convex MINLP problems demonstrate that the proposed algorithm produces tighter MILP relaxations than the classical outer approximation method. Furthermore, the approach offers an alternative mechanism for generating linear cuts, extending beyond reliance solely on first-order Taylor expansions and shows that the efficiency of outer approximation algorithm is strongly dependent on the inherent structure of the MINLP problem.

New Outer Approximation Algorithms for Nonsmooth Convex MINLP Problems

TL;DR

The paper addresses solving nonsmooth convex MINLPs by introducing an outer-approximation framework that reformulates the problem as an equivalent MILP master and solves a sequence of MILP relaxations with finite termination. It leverages KKT-based subgradients and a new parameter derived from nonlinear constraints to generate stronger cuts, ensuring equivalence between the MILP master and the original problem . The method yields tighter relaxations than classical OA and demonstrates computational efficiency gains by exploiting problem structure rather than relying solely on first-order Taylor expansions. The resulting approach has practical impact for a broad class of MINLPs, offering an effective alternative to existing OA methods and enhancing solution quality and speed in convex, nonsmooth settings.

Abstract

This paper presents a novel outer approximation algorithm for nonsmooth mixed-integer nonlinear programming (MINLP) problems. The method proceeds by fixing the integer variables and solving the resulting nonlinear convex subproblem. When the subproblem is feasible, valid linear cuts are derived by computing suitable subgradients of the objective and constraint functions at the optimal solution, utilizing KKT optimality conditions. A new parameter, defined through the nonlinear constraint functions, is introduced to facilitate the generation of these cuts. For infeasible subproblems, a feasibility problem is solved, and valid linear cuts are generated via KKT-based subgradients to exclude the infeasible integer assignment. By integrating both types of cuts, a mixed-integer linear programming (MILP) master problem is formulated and proven equivalent to the original MINLP. This equivalence underpins a new outer approximation algorithm, which is guaranteed to terminate after a finite number of iterations. Numerical experiments on smooth convex MINLP problems demonstrate that the proposed algorithm produces tighter MILP relaxations than the classical outer approximation method. Furthermore, the approach offers an alternative mechanism for generating linear cuts, extending beyond reliance solely on first-order Taylor expansions and shows that the efficiency of outer approximation algorithm is strongly dependent on the inherent structure of the MINLP problem.
Paper Structure (12 sections, 9 theorems, 57 equations, 6 figures)

This paper contains 12 sections, 9 theorems, 57 equations, 6 figures.

Table of Contents

  1. Introduction
  2. MINLP Problems and Solution Methods
  3. Related Research on OA in the Literature
  4. Contributions and Outline of the Paper
  5. Preliminaries
  6. New OA Algorithm for Convex MINLP Problems
  7. Feasible Nonlinear Subproblem.
  8. Infeasible Nonlinear Subproblem.
  9. Construction of New OA Algorithm
  10. Compared with Classic OA for Convex MINLP
  11. Numerical results
  12. Concluding remarks

Key Result

Lemma 2..1

Let $\phi:\mathbb{R}^n\times\mathbb{R}^p\rightarrow \mathbb{R}$ be a continuous and convex function and suppose that $(\bar{x}, \bar{y})\in \mathbb{R}^n\times\mathbb{R}^p$. Then for any $\alpha\in\partial\phi(\cdot,\bar{y})(\bar{x})$, there exists $\beta\in\mathbb{R}^p$ such that $(\alpha, \beta)\in

Figures (6)

  • Figure 1: The figure to the left shows the feasible regions defined by the individual constraints of problem \ref{['3.36-251216']}. The right figure shows the integer relaxed feasible region (i.e., the region without considering integer restrictions), contours of the objective function, the optimal solution ($\bigstar$) of the problem, and the initial point($\bigstar$).
  • Figure 2: Applying the OA algorithm to problem \ref{['3.36-251216']} results in multiple iterations, and the progress of the algorithm is shown, with each figure being an iteration. The figures show the feasible region defined by the nonlinear constraints in dark gray, and the light gray region shows the outer approximation obtained by the generated cuts. The squared dots ($\blacksquare$) represent the solution obtained from the MILP master problem, and the circular dots ($\bullet$) represent the solution obtained by the NLP subproblem.
  • Figure 3: The figure demonstrates the iterative process of solving problem \ref{['3.36-251216']} using the New OA algorithm, and it adopts the format of Fig \ref{['fig:2']}.
  • Figure 4: The left figure shows a cutting line () generated by the OA algorithm during the second iteration. The right figure shows a cutting line () produced by the new OA algorithm during the second iteration, where $\rho_2 = 2.3464 / 12.8289 \approx 0.1829$. The yellow square points($\blacksquare$) in the figure represent feasible points when $y = 13$, and the green square points($\blacksquare$) represent feasible points when $y = 15$. The purple star($\bigstar$) represents the optimal solution to this problem. The format from Figure 2 is used here.
  • Figure 5: Bound profiles of the upper bound (UB) and lower bound (LB) over computation time for instance cvxnonsep_normcon30, solved by the OA and New OA optimization algorithms.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 2..1
  • Lemma 2..2
  • Lemma 2..3
  • Proposition 3..1
  • Proposition 3..2
  • Remark 3..1
  • Theorem 3..1
  • Remark 3..2
  • Proposition 3..3
  • Theorem 3..2
  • ...and 2 more