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A condensing approach for linear-quadratic optimization with geometric constraints

Alberto De Marchi

TL;DR

This work addresses linear-quadratic optimization with geometric constraints that may be nonconvex but projection-friendly. It extends the augmented Lagrangian framework by condensing the $x$-subproblem into a marginal function over $z$, exploiting the SPD structure to precondition and reduce problem size, while treating hard lower-level equalities when available. The resulting method achieves substantial computational gains across diverse control benchmarks and maintains convergence guarantees under mild assumptions. Practically, the approach enables fast, scalable solutions for complex constraint sets encountered in signal processing, control, and decision-making tasks.

Abstract

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and cardinality constraints, among others. In particular, we cover also situations where parts of the constraints are nonconvex and possibly complicated, but it is practical to compute projections onto this nonconvex set. Our approach combines the augmented Lagrangian framework with a solver-agnostic structure-exploiting subproblem reformulation. While convergence guarantees follow from the former, the proposed condensing technique leads to significant improvements in computational performance.

A condensing approach for linear-quadratic optimization with geometric constraints

TL;DR

This work addresses linear-quadratic optimization with geometric constraints that may be nonconvex but projection-friendly. It extends the augmented Lagrangian framework by condensing the -subproblem into a marginal function over , exploiting the SPD structure to precondition and reduce problem size, while treating hard lower-level equalities when available. The resulting method achieves substantial computational gains across diverse control benchmarks and maintains convergence guarantees under mild assumptions. Practically, the approach enables fast, scalable solutions for complex constraint sets encountered in signal processing, control, and decision-making tasks.

Abstract

Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and cardinality constraints, among others. In particular, we cover also situations where parts of the constraints are nonconvex and possibly complicated, but it is practical to compute projections onto this nonconvex set. Our approach combines the augmented Lagrangian framework with a solver-agnostic structure-exploiting subproblem reformulation. While convergence guarantees follow from the former, the proposed condensing technique leads to significant improvements in computational performance.
Paper Structure (19 sections, 2 theorems, 31 equations, 5 figures)

This paper contains 19 sections, 2 theorems, 31 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Optimality and Stationarity
  3. Augmented Lagrangian Method
  4. Safeguarded augmented Lagrangian scheme
  5. Convergence and infeasibility detection
  6. Inner solver: projected gradient
  7. Condensing: How and Why
  8. Evaluating the marginal function
  9. Lower-level equality constraints
  10. Numerical Results
  11. Initial value problem
  12. Obstacle problem
  13. AFTI-16 tracking control
  14. Conclusions
  15. Additional Material
  16. ...and 4 more sections

Key Result

Proposition 3.1

Suppose matrix $A$ in eq:P has full rank and the sequence $\{\varepsilon_k\}$ in alg:ALM satisfies $\varepsilon_k\to 0$. Then each feasible accumulation point $(\bar{x},\bar{z})$ of a sequence $\{(x_k,z_k)\}$ generated by alg:ALM is a KKT point of eq:Pz. In particular, each feasible accumulation poi

Figures (5)

  • Figure 1: Initial value problem \ref{['eq:initialValueProblem']}: data (top) and performance (bottom) profiles for comparing different solver settings.
  • Figure 2: Obstacle problem \ref{['eq:obstacleProblem']}: data (top) and performance (bottom) profiles for comparing different solver settings. Legend as in \ref{['fig:ivp']}.
  • Figure 3: AFTI-16 problem \ref{['eq:initialValueProblem']}: data (top) and performance (bottom) profiles for comparing different solver settings. Legend as in \ref{['fig:ivp']}.
  • Figure 4: Closed-loop simulation of AFTI-16 behavior: outputs (top), inputs (middle) and solver runtime (bottom) using cold- (black) or warm-starting (orange) between consecutive simulation steps. A disturbance is injected at half time, resetting the observed states as at the initial time.
  • Figure 5: Scalability profiles for comparing different solver settings: initial value problem \ref{['eq:initialValueProblem']} (top), obstacle problem \ref{['eq:obstacleProblem']} (middle) and AFTI-16 problem \ref{['eq:afti16problem']} (bottom). Legend as in \ref{['fig:ivp']}.

Theorems & Definitions (2)

  • Proposition 3.1
  • Proposition 3.2