PaperlayerView exact revision on arXiv

Source-linked AI summary

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Kunal Garg, Dimitra Panagou

arXiv:1808.10474v5math.OC

TL;DR

Continuous-time optimization has largely focused on asymptotic or exponential convergence, while fixed-time guarantees are independent of initial conditions. This paper develops modified gradient flows, a Newton-based method, and saddle-point schemes for several optimization settings, and reports faster iteration counts and lower wall-clock time than nominal Newton and rescaled-gradient methods in its numerical comparisons.

  • Problem

    Most continuous-time optimization work addresses asymptotic or exponential convergence, whereas this paper targets convergence within a fixed time independent of initial conditions.

    Source: A. Motivation · p. 1

  • Method

    The paper develops modified gradient-flow and Newton-based schemes for unconstrained and equality-constrained optimization, alongside modified saddle-point dynamics for min-max problems.

    Source: E. Contributions of the paper · p. 2, §VII · p. 13

  • Results

    The proposed method achieves faster convergence in iteration count and lower wall-clock time than nominal Newton's method and the rescaled-gradient method in the reported numerical comparisons.

    Source: E. Contributions of the paper · p. 3

  • Takeaways & Limitations

    The numerical illustrations support the practical applicability of the proposed methods across unconstrained, constrained, and min-max optimization examples.

    Source: E. Contributions of the paper · p. 3

    The paper leaves discretization schemes that preserve finite- or fixed-time convergence, and guarantees in a finite or fixed number of steps, for future work.

    Source: §VI · p. 13

Abstract

from arXiv · show

This paper proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a \textit{fixed} time from any given initial condition for unconstrained optimization, constrained optimization, and min-max problems. The application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient-dominance. Then, a modified Newton's method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Building upon this method, a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed. More specifically, constrained optimization problems formulated as min-max problems are considered, and a novel method for computing the optimal solution in fixed-time is proposed using the Lagrangian dual. Finally, the general min-max problem is considered, and a modified scheme to obtain the optimal solution of saddle-point dynamics in fixed time is developed. Numerical illustrations that compare the performance of the proposed method against Newton's method, rescaled-gradient method, and Nesterov's accelerated method are included to corroborate the efficacy and applicability of the modified gradient flows in constrained and unconstrained optimization problems.

I. INTRODUCTION

The paper develops continuous-time optimization schemes with fixed-time convergence, addressing a literature that had largely focused on asymptotic or exponential convergence. Its contributions span unconstrained, equality-constrained, and min-max problems, with numerical illustrations of performance.

A. Motivation

  • The proposed methods target convergence within a bound independent of initial conditions, beyond the asymptotic or exponential convergence common in prior continuous-time optimization work.Finite-time convergence can have a settling time that grows with the initial condition.Source: A. Motivation · p. 1, C. Finite-time and fixed-time stability · p. 2

B. Gradient flows: theory and applications

  • Gradient flows are used across unconstrained, constrained, and saddle-point optimization, and Lyapunov stability theory provides convergence proofs for their continuous-time dynamics.Examples of saddle-point applications include network optimization and zero-sum games.Source: B. Gradient flows: theory and… · pp. 1–2

C. Finite-time and fixed-time stability

  • Fixed-time stability strengthens finite-time stability by uniformly bounding the settling time across all initial conditions.The paper also reviews prior results on robustness of finite-time and fixed-time equilibria under disturbances.Source: C. Finite-time and fixed-time stability · p. 2

D. Related prior work on continuous-time optimization

  • Unlike several earlier finite-time schemes with discontinuous dynamics, the paper designs continuous modified flows and proves existence and uniqueness for all times and initial conditions.The paper also contrasts its fixed-time guarantees with finite-time approaches whose settling time grows with initial conditions.Source: D. Related prior work on… · p. 2

E. Contributions of the paper

  • The paper proposes modified gradient flows for unconstrained and constrained optimization, plus min-max problems, with fixed-time convergence guarantees.The work also studies saddle-point dynamics and constrained optimization through min-max formulations.Source: A. Motivation · p. 1, E. Contributions of the paper · p. 2
  • The contributions include a gradient-dominance-based unconstrained flow, a Newton-based method for strictly convex objectives, equality-constrained optimization under stated conditions, and modified saddle-point dynamics.The equality-constrained method uses a closed-form conjugate; the paper also describes a saddle-point approach for constrained problems when the conjugate is unavailable in closed form.Source: E. Contributions of the paper · p. 2
  • Numerical illustrations report faster convergence in iteration count and lower wall-clock time than nominal Newton's method and the rescaled-gradient method.The paper includes examples involving a support-vector machine, a quadratic program, and a min-max problem.Source: E. Contributions of the paper · pp. 2–3, F. Organization · p. 3

F. Organization

  • The paper is organized around stability and convex-optimization preliminaries, fixed-time methods, numerical examples, discussion, and future work.The numerical examples cover a support-vector machine, a quadratic program, and a min-max problem.Source: F. Organization · p. 3

A. Notation

This section sets notation and reviews stability and convexity concepts used in the paper. It distinguishes finite-time convergence, whose settling time may depend on the initial condition, from fixed-time convergence, which has a uniform bound.

A. Notation

  • The notation defines the optimal value and point, the conjugate function, gradient and Hessian symbols, differentiability classes, and matrix definiteness conventions.The paper distinguishes the optimal-value marker ⋆ from the conjugate marker ∗.Source: A. Notation · p. 3

B. Mathematical preliminaries

  • Finite-time stability requires convergence in finite time, while fixed-time stability additionally bounds the settling time uniformly over all initial conditions.Ordinary finite-time settling time can grow unbounded as the initial state moves farther away.Source: B. Mathematical preliminaries · p. 3
  • The preliminaries summarize convexity, concavity, strict and strong convexity, strong smoothness, and first- and second-order characterizations.Strong convexity implies strict convexity, which implies convexity.Source: B. Mathematical preliminaries · p. 4
  • The section also defines locally convex-concave functions by requiring convexity in one variable and concavity in the other, locally around the points considered.The definition also covers locally strongly and locally strictly convex-concave functions.Source: B. Mathematical preliminaries · p. 4

III. FXTS IN OPTIMIZATION

The section introduces gradient-flow schemes for unconstrained and constrained convex optimization, beginning with a rescaled flow for unconstrained problems. Under strong convexity, that flow reaches the unique optimum in finite time, although its settling-time bound depends on the initial condition.

III. FXTS IN OPTIMIZATION

  • The section proposes novel gradient-flow schemes for unconstrained and constrained convex optimization, then first treats unconstrained optimization.Source: §III · p. 4

A. Unconstrained optimization: FTS scheme

  • The rescaled gradient flow is established as a finite-time method under strong convexity, with bounded sublevel sets supporting convergence from every initial condition.Strong convexity makes the optimum unique, and the proof uses a Lyapunov function based on the gradient norm.Source: A. Unconstrained optimization: FTS… · p. 4
  • For a C2 strongly convex objective, the rescaled gradient flow converges to the unique optimum in finite time for any p>2.The finite settling-time bound depends on the initial condition and grows with distance from the optimum.Source: A. Unconstrained optimization: FTS… · pp. 4–5
  • The settling-time bound for this finite-time scheme grows with the initial distance, motivating a modified flow designed to reach the optimum within a fixed time.The paper assumes the objective attains its minimum; coercivity is given as a sufficient condition for existence of a minimizer.Source: A. Unconstrained optimization: FTS… · pp. 4–5

B. Unconstrained optimization: FxTS-GF scheme

The FxTS-GF scheme gives fixed-time convergence for unconstrained minimization under gradient dominance and a unique minimizer. Its additional gradient term supports faster convergence for both small and large distances, and the framework includes objectives that satisfy PL without being strongly convex.

B. Unconstrained optimization: FxTS-GF scheme

  • The flow's equilibria are exactly the critical points, and under the stated regularity and convexity conditions its solutions exist uniquely for all forward times.The paper's proof uses local Lipschitz continuity of the gradient and the specified exponent conditions.Source: B. Unconstrained optimization: FxTS-GF… · p. 5
  • Under a unique minimizer and the PL inequality, the FxTS-GF trajectories converge to the optimum in fixed time from every initial condition.The paper assumes the minimum is attained; it notes that convexity is not required by the PL condition itself.Source: B. Unconstrained optimization: FxTS-GF… · p. 5
  • The added term with exponent 1<p2<2 complements the p1>2 term, producing convergence terms that dominate at small and large distances, respectively.The comparison with rescaled gradient flow explains that the additional term addresses the large-initial-distance slowdown of that flow.Source: B. Unconstrained optimization: FxTS-GF… · p. 6
  • The FxTS-GF applies to objectives satisfying PL even when they are not strongly convex, including least-squares and linear-regression forms under the stated uniqueness condition.For compositions involving a matrix A, the paper states that the resulting objective can satisfy PL even when A is not full row-rank.Source: B. Unconstrained optimization: FxTS-GF… · p. 6

C. FxTS Newton’s method

The section modifies Newton’s method to obtain fixed-time convergence for strictly convex objectives, including cases that need not satisfy the Polyak–Łojasiewicz inequality. A quartic reformulation also brings sign-constrained quadratic programs into this framework.

C. FxTS Newton’s method

  • The section presents the modified Newton method as a fixed-time-convergent alternative for optimization, with the nominal Newton method serving as the comparison point.The passage introduces the nominal Newton method but supplies no comparative performance result here.Source: C. FxTS Newton’s method · p. 6
  • Under Assumptions 1 and 3, the modified Newton flow converges to the unique optimum in fixed time for every initial condition.The objective must be strictly convex; the flow uses parameters p1 > 2 and 1 < p2 < 2.Source: C. FxTS Newton’s method · p. 6
  • Strictly convex objectives that do not satisfy the PL inequality can still be solved in fixed time with the modified Newton method.Strong convexity implies the PL inequality, but strict convexity alone does not; the paper identifies quartic functions as an example class.Source: C. FxTS Newton’s method · p. 6
  • The sign-constrained QP is reformulated as an unconstrained quartic by substituting xi = zi^2, allowing the modified Newton method to find its optimum in fixed time.The resulting objective is strictly convex but may be neither strongly convex nor PL; the optimum is recovered using x̄i = z̄i^2.Source: C. FxTS Newton’s method · p. 7

D. Convex optimization with linear equality constraints

For convex optimization with linear equality constraints, the method first solves a dual problem and then minimizes the Lagrangian at the dual optimum. Under the stated assumptions, this two-stage procedure obtains the constrained optimum in fixed time.

D. Convex optimization with linear equality constraints

  • The framework assumes a convex objective, a full-row-rank constraint matrix, and a coercive objective so the constrained optimum exists.The paper notes that full row rank makes the feasible set nonempty and closed, while coercivity ensures existence.Source: D. Convex optimization with linear… · p. 7
  • Strong smoothness of f makes the dual function strongly concave, supporting fixed-time convergence of the dual flow to ν⋆.The stated assumptions also require a known closed-form conjugate f* with specified regularity.Source: D. Convex optimization with linear… · p. 7
  • After finding ν⋆, the Lagrangian minimizer is the constrained optimizer, and the corresponding primal flow converges to it.The paper identifies the optimum through ∇xL(x⋆, ν⋆) = 0.Source: D. Convex optimization with linear… · p. 7
  • The constrained optimum is obtained in fixed time by first solving the dual flow for ν⋆ and then the primal flow for x⋆.The total settling time is bounded by the sum of the dual and primal settling times.Source: D. Convex optimization with linear… · pp. 7–8
  • Equality-constrained quadratic programs with positive-definite Q and full-row-rank A fit the proposed framework.The paper states that the associated objective and conjugate satisfy Assumption 5.Source: D. Convex optimization with linear… · p. 8
  • The gradient regularity assumption can be relaxed: continuity of the vector field suffices for solution existence and uniqueness, and strongly convex f needs no additional assumptions on f*.The remark states that f ∈ C1 suffices in the strongly convex case.Source: D. Convex optimization with linear… · p. 8

IV. FXTS OF SADDLE-POINT DYNAMICS

The modified FxTS saddle-point dynamics bring min-max and linear equality-constrained optimization problems to their saddle points in fixed time. The guarantees depend on the stated convexity, Hessian, and initial-condition assumptions, with a first-order alternative available under stronger conditions.

IV. FXTS OF SADDLE-POINT DYNAMICS

  • Dynamics (30) modify the saddle-point flow used in with the FxTS Newton method; strict convexity-concavity makes its equilibrium the unique saddle point.The method uses positive c1 and c2 with p1 > 2 and 1 < p2 < 2.Source: §IV · p. 8
  • Under Assumption 6, the trajectories of (30) reach the saddle point in fixed time for initial conditions in D, and globally when U = R^n × R^m.D is the largest compact sublevel set specified for the theorem.Source: §IV · pp. 8–9
  • Corollary 1 gives fixed-time convergence to critical points without strict convexity-concavity, and to the saddle point locally when it is the only critical point in the set.The stated sufficient Hessian conditions are invertibility of ∇xxF and full row-rank of ∇zxF.Source: §IV · p. 9
  • For linear equality-constrained problems, taking F(x, z) = f(x) + z^T(Ax − b) yields the optimizer as a saddle point in fixed time under Assumption 4 and strict convexity of f.The cited remark also notes this formulation can be used when the conjugate function is not known in closed form.Source: §IV · p. 9
  • The paper reports fixed-time guarantees for constrained and general min-max problems under relaxed assumptions compared with the asymptotic-convergence results in,.Source: §IV · p. 9
  • A first-order saddle-point alternative requires only ∇F but assumes local strong convexity-concavity, and its trajectories converge to the saddle point in fixed time.The theorem gives the result for initial conditions in U, and globally when U = R^n × R^m.Source: §IV · p. 9

V. NUMERICAL EXAMPLES

The numerical section uses three examples to illustrate the proposed methods on unconstrained, equality-constrained, and min-max problems. Implementations use MATLAB and, unless otherwise stated, Euler discretization with a fixed time step.

V. NUMERICAL EXAMPLES

  • Unless otherwise stated, simulations use MATLAB R2018a, Euler discretization, and a constant time step of 10^-5 seconds.The paper notes that a convergence time T in seconds corresponds to T × 10^5 iterations under this step size.Source: §V · p. 10
  • The first example compares FxTS-GF with Newton’s method on logistic-regression SVM, while the second applies FxTS saddle-point dynamics to a quadratic program with equality constraints.Source: §V · p. 10
  • The third example applies FxTS saddle-point dynamics to an instance of a general min-max problem.Source: §V · p. 10

A. Support Vector Machine: Unconstrained optimization

The logistic SVM example uses randomly distributed labeled points and compares the proposed FxTS-GF with nominal gradient flow. The reported comparison emphasizes convergence speed across initial conditions.

A. Support Vector Machine: Unconstrained optimization

  • The example uses 500 randomly distributed labeled points around x1 = x2, with the separating hyperplane solution x⋆ = [1, −1].The labels li take values −1 and 1.Source: A. Support Vector Machine:… · p. 10, Figure 1 · p. 10
  • At least 5 times fewer iterations are reported for FxTS-GF than nominal GF to reach the same accuracy, with convergence to 10^-10 within the theoretical bound T1.The stated bound is T1 ≤ 0.43 seconds.Source: A. Support Vector Machine:… · p. 10
  • The plotted FxTS-GF trajectories show super-linear convergence, while the nominal gradient-flow plots are linear on the log scale, indicating exponential convergence.Source: A. Support Vector Machine:… · p. 10

B. Example 2: QP with equality constraints

The equality-constrained quadratic-program example applies FxTS saddle-point dynamics and compares its convergence across initial conditions with nominal saddle-point dynamics. The reported error threshold is reached within the stated fixed-time bound.

B. Example 2: QP with equality constraints

C. Example 3: Min-max problem

In this min-max example, the proposed scheme’s convergence time remains bounded across initial conditions, and its convergence rate changes with the parameters p1 and p2. The experiments also compare discretization steps and the proposed scheme with rescaled-gradient flow.

C. Example 3: Min-max problem

VI. DISCUSSION

The paper reports fixed-time continuous-time optimization schemes whose discretized implementations perform well numerically, while proving that discretization preserves their guarantees remains open. It identifies broader constraint classes, function-space problems, and guarantee-preserving discretization as future directions.

VI. DISCUSSION

  • Numerical discretizations perform consistently with the theoretical results, but proving that discretization preserves fixed-time guarantees remains open.The discussion notes prior consistent-discretization results for particular homogeneous systems and practically fixed-time-stable systems, while leaving extensions to the paper’s methods for future work.Source: §VI · p. 13

VII. CONCLUSIONS AND FUTURE WORK

  • The proposed schemes achieve super-linear convergence in discretized examples within the theoretical time bound and outperform Newton’s and rescaled-gradient methods.The paper presents modified gradient-flow schemes under strict-convexity or gradient-dominance assumptions, a modified Newton method, and fixed-time methods for equality-constrained and min-max problems.Source: §VII · p. 13
  • Future work includes fixed-time methods for general equality-and-inequality-constrained problems and optimization over function spaces.The authors also identify discretization schemes that preserve convergence time and translate continuous-time guarantees into finite- or fixed-step convergence.Source: §VII · p. 13

APPENDIX A PROOF OF LEMMA 5

The appendices establish equilibrium, continuity, dual-concavity, and Hessian-invertibility properties under the stated conditions. These results support the paper’s fixed-time optimization analyses.

APPENDIX A PROOF OF LEMMA 5

  • For p1 > 2 and 1 < p2 < 2, an equilibrium of dynamics (8) is characterized by ∇f(x) = 0.The proof establishes that x = x̄ is an equilibrium if and only if the gradient vanishes at x̄.Source: Appendix A · p. 14

APPENDIX B PROOF OF LEMMA 6

  • If f is C1,1_loc, the right-hand side of dynamics (8) is continuous, including at points where ∇f vanishes.The proof uses local Lipschitz continuity of ∇f near each such point and the equilibrium characterization from Lemma 5.Source: Appendix B · pp. 14–15

APPENDIX C PROOF OF LEMMA 7

  • With convex, strongly smooth f and full-row-rank A, the Lagrangian dual function g is strongly concave.The proof obtains strong concavity from the strong convexity of f*(−A^Tν), using the positive minimum eigenvalue of AA^T.Source: Appendix C · p. 15

APPENDIX D PROOF OF LEMMA 9

  • Under Assumption 6, positive-definite Hxx and Hzz imply that the Hessian ∇²F(x, z) is invertible on U.The proof establishes full rank by decomposing the Hessian rank into the ranks associated with Hxx and the corresponding Schur-complement block.Source: Appendix D · p. 15
Loading 1808.10474v5…