Optimization with Equality and Inequality Constraints Using Parameter Continuation

Harry Dankowicz; Mingwu Li

REVIEW 3 major objections 4 minor 24 references

Optimization with Equality and Inequality Constraints Using Parameter Continuation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Staged continuation reaches constrained optima from infeasible starts

desk verdict A genuine extension of the Kernévez–Doedel successive continuation framework to inequality constraints, with honest numerical demonstrations and a load-bearing reachability assumption that the paper itself concedes. read the letter →

arxiv 1909.01422 v1 pith:OB2HQVXR submitted 2019-09-03 math.OC math.DS

Mingwu Li , Harry Dankowicz This is my paper · ORCID

classification math.OCmath.DS MSC 49K1549K2749M0549M2990C3345J0534B99

keywords constrainedoptimizationcomplementarityconditionsKarush-Kuhn-Tuckersuccessivecontinuationboundary-valueproblemsoptimalcontrolFischer-Burmeisterfunctionparameter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that local optima of an objective subject to both equality and inequality constraints can be found by a sequence of easy-to-initialize continuation runs, without ever needing a starting guess for the Lagrange multipliers. The trick is to replace the complementary slackness conditions of the Karush-Kuhn-Tucker theory by relaxed equations built from a complementarity function, then drive the relaxation parameters to zero through successive stages of continuation. The paper proves lemmas showing how branch points in these stages allow nonzero multipliers to appear from a zero-multiplier start, and it shows that the initial guess may even violate the inequality constraints. If true, this removes the main practical barrier--finding an adequate initial guess for the multipliers--for optimization along families of boundary-value problems and in optimal control.

What carries the argument

The engine of the method is the augmented system $F_{\mathrm{aug}}=0$ built from (i) the equality constraints $\Phi(u)=0$ and objective monitor $\Psi(u)-\mu=0$, (ii) the adjoint condition $(D\Phi(u))^*\lambda+(D\Psi(u))^*\eta+(DG(u))^*\sigma=0$ with normalization $\eta-\nu=0$, and (iii) the relaxed complementary conditions $\chi(\sigma_i,-G_i(u))-\kappa_i=0$ for each inequality. The complementarity function $\chi(a,b)=\sqrt{a^2+b^2}-a-b$ vanishes exactly when $a,b\ge 0$ and $ab=0$, so the equations enforce complementary slackness in the limit $\kappa=0$; its nonsmooth zero contour at the origin is what creates the singular points where branch switching to nonzero multipliers can occur. The continuation parameters $\nu$ and $\kappa$ are frozen or released stage by stage, and the lemmas show when the corresponding solution sets are one-dimensional manifolds, when a stationary point of the objective is a branch point, and when a secondary branch proceeds to the desired terminal values.

What would settle it

A concrete check: take a smooth constrained problem with a unique KKT point and try every admissible initial point and index order; if every staged path either stalls at a negatively aligned singular point or leaves the computational domain before $\nu=1$ and $\kappa=0$ are reached, the claimed search-from-infeasible-start property fails. The paper's own motivating example shows such a stall for one initial choice ($(3,1)$ stalling on $G_2=0$), so the question is whether any problem exhibits this stall for all choices.

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Extended reading notes

Core claim

The central claim is that a generalized successive continuation paradigm solves constrained optimization problems with simultaneous equality and inequality constraints by seeking roots of an augmented system that includes the original equations, adjoint conditions linear and homogeneous in the Lagrange multipliers, and relaxed complementarity conditions of the form $\chi(\sigma_i,-G_i(u))=\kappa_i$, with $\chi(a,b)=\sqrt{a^2+b^2}-a-b$. Starting from a solution of the equality constraints with all multipliers set to zero and with $\kappa_i=0$ for inactive inequalities and $\kappa_i>0$ for violated ones, the algorithm follows one-dimensional solution manifolds, detects fold points or singular points, switches branches to manifolds on which multipliers become nonzero, and finally drives the relaxation parameters $\kappa_i$ (and normalization parameters $\nu$) to their KKT limits. The paper's lemmas establish, under rank conditions, that the necessary branch points exist and that the terminal point of the sequence satisfies the KKT conditions. Consequently, the search can begin from a potentially infeasible initial guess and requires no nonzero multiplier initialization.

Load-bearing premise

Each continuation stage must actually be able to drive its target parameter ($\nu$ components to 1, $\kappa$ components to 0) within the chosen computational domain, which may require successfully bypassing singular points; the paper does not prove this reachability for all problems.

Editorial extensions

If this is right

For a problem with $d$ unknown design or control variables, the staged construction reaches a KKT point in $d+1$ continuation runs (or $d$ runs when the first stage uses branch switching), independent of the number of constraints.
Zero Lagrange multipliers and an infeasible initial point are enough to start the search; the method does not require a feasible starting guess.
Inequality constraints add new opportunities for branch switching, so they can help rather than merely obstruct the continuation search.
The formulation is compatible with staged-construction continuation software that generates adjoints automatically, so it applies to two-point boundary-value problems and optimal control problems with integral inequality constraints.
If a secondary branch is negatively aligned at a singular point, the standard predictor-corrector continuation algorithm may fail there; the paper notes that manual switching or an alternative initial choice and ordering of stages is then needed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural test is to replace the nonsmooth complementarity function by a smooth approximant and add a final continuation stage that drives the smoothing parameter to zero; this could eliminate many of the singular points that force manual branch switching, an extension the paper itself raises.
Because the order in which components of $\nu$ and $\kappa$ are driven to their limits is chosen by the user, different orders may converge to different local optima; a systematic ordering rule could be benchmarked on multi-extremum problems.
The framework is stated for finite-dimensional inequality constraints, so a meaningful extension is to discretize infinite-dimensional inequality constraints in a way consistent with the adjoint discretization and test whether the staged convergence still reaches KKT points.
The same complementarity relaxation could be applied to regularize bang-bang optimal control problems, treating the singular limit as the terminal continuation stage; the paper mentions such regularization as a possible combination.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is the short version. This is a real extension of the Kernévez–Doedel successive continuation idea to inequality constraints, and the numerical demonstrations support the claim that infeasible starts can work. The central caveat is that the paper's own examples show the staged scheme can fail, and Section 7 concedes the reachability assumption. That assumption is load-bearing, so the headline claim should be read as "can succeed" rather than "is guaranteed."

What is new: the Fischer–Burmeister complementarity function inside successive continuation, the branch-point lemmas for inequality constraints, and the demonstration of infeasible-start searches for KKT points in BVP-constrained and optimal control problems. The finite-dimensional example is worked analytically and matches the continuation results, which is real evidence. The appendix lemmas give a local picture: branch points exist and secondary branches are locally parameterized by eta_1. That is useful.

The soft spots are real but not fatal. The lemmas are sketches. They rely on "generically" and continuity arguments, and they do not provide checkable conditions for positive branch alignment or for driving nu and kappa to their target values. The stress-test note is right: Lemma A.6 only gives a local manifold on the G_k = 0 surface, not the alignment needed for the pseudo-arclength algorithm to bypass the singular point. The paper itself reports cases where continuation terminates at singular points or domain boundaries and manual restart is needed. That is not fatal for a methods paper, but it means the algorithm is a heuristic with theoretical scaffolding, not a provable method. Also, the modified COCO core is not released, so the BVP and optimal-control results are not immediately reproducible. That should be fixed before publication.

The citation pattern is fine. Self-citation to the authors' earlier COCO work is appropriate because that is the implementation the paper builds on; there is no circular fitting.

Who this is for: people working on numerical continuation for constrained optimization, especially in COCO, will get a clear framework and worked examples. I would send this to a serious referee. The right outcome is probably major revision, not desk rejection, with referees asking for clearer assumptions and code or data release. The core idea holds up.

Referee Report

3 major / 4 minor

Summary. The paper generalizes the successive continuation paradigm of Kernévez and Doedel to constrained optimization problems with simultaneous equality and inequality constraints. The authors augment the KKT conditions with the Fischer-Burmeister complementarity function, define a family of restricted continuation problems, and use staged continuation to drive relaxation parameters to their limiting values. They prove several local lemmas (Appendix A), implement the method by extending the COCO software package, and demonstrate it on a finite-dimensional example, a two-point boundary-value problem with an integral inequality constraint, and an optimal control problem. The central claim is that optima can be found from easily initialized stages, without seeding nonzero Lagrange multipliers in the first stage, and starting from points that may violate the inequality constraints.

Significance. If the central claim is fully established, the paper offers a genuinely useful extension of a known continuation-based optimization technique, with potential applications to boundary-value and optimal control problems. The authors provide analytic and numerical evidence for the finite-dimensional motivating example, a working software implementation, and explicit discussion of the method's limitations, which is commendable. The local lemmas in Appendix A are plausible and provide a useful framework for understanding branch points and secondary branches. However, the paper's headline claim about a successful search from infeasible initial guesses is not fully supported by the theoretical analysis, because the global reachability of each continuation stage is assumed rather than proved, and the numerical examples themselves reveal multiple failures that require manual intervention or fortuitous branch alignment.

major comments (3)

[Section 7 and Sections 3, 6.1] The paper's own concluding section admits that the success of the method depends on an unproved assumption: 'It has been tacitly assumed that each successive stage of continuation is able to drive the appropriate continuation parameters to their desired values, preferably monotonically. But the examples showed that this may not be possible within a given computational domain, or may only be possible by occasionally bearing in a direction away from the desired values.' This is a load-bearing assumption for the abstract's claim that 'a successful search for optima is found to be possible also from an infeasible initial solution guess,' because every stage of the algorithm requires reaching a target value of a component of ν or κ. The finite-dimensional example in Section 3 explicitly fails for the initial point u0 in U+/-: continuation terminates at the singular point (7/5, 7/5) on G2=0, the secondary branch is negatively aligned, and the authors restart from a different infeasible point. Similarly, in Section 6.1 the run starting from FP2 fails to reach νp2=0, terminating at a singular point on G_int=0, and the authors note that success would require a larger computational domain and continuation in the other direction. These failures are not isolated artifacts: they concern the central mechanism by which the method is supposed to reach a KKT point.
[Appendix A, Lemma A.6 and Section 4, paragraph 5] Lemma A.6 establishes only a local one-dimensional manifold on the zero-level surface of G_k, with σ_k and σ_P nonzero for η1 close to 0. It does not provide any criterion ensuring that the secondary branch can be followed to η1=1, nor that the tangent directions at the singular point are positively aligned so that the pseudo-arclength algorithm can automatically bypass the singularity. The text in Section 4 states 'Suppose that no element of G_Z equals 0 along this manifold for η1∈[0,1]' and 'Lemma A.6 allows for the possibility of branch switching,' but these suppositions are not derived from the lemmas. The example in Section 3 shows exactly the opposite: at the singular point on G2=0 the secondary branch is negatively aligned and the pseudo-arclength algorithm cannot bypass it. Without a global reachability or alignment condition, the proposed procedure remains a heuristic with local justification, not a rigorous guarantee of successful search.
[Section 4, 'Initialization' bullet and Section 7, first paragraph] The paper describes a 'rigorous framework' in the conclusions, but the lemmas in Appendix A are proofs by sketch relying on 'generically' and continuity arguments (e.g., Lemma A.3), and they do not quantify how far the secondary branches can be continued. Moreover, the algorithm requires the user to choose the initial point u0, the index sets I and P, the continuation order, and the computational domain; the paper itself states 'We are not able to propose a systematic selection algorithm beyond the principles outlined above.' While user choices are common in continuation methods, the combination of an unproved reachability assumption and the acknowledged need for manual switching or restarts in several examples means the manuscript overstates the degree to which it delivers a reliable search method. The claims should either be weakened to describe the method as a set of locally justified heuristics, or supplemented with additional conditions under which the staged continuation is guaranteed to reach a KKT point.

minor comments (4)

[Section 2, Eq. (2.7)-(2.10)] The singularity of the Fischer-Burmeister function at (0,0) is identified, and the paper correctly assumes that no active inequality constraints are present at the initial point in the lemmas. The notation κ0 and the role of the set P would benefit from being summarized in a table for readability, but this is not essential.
[Section 3, numerical results paragraphs] The numerical results for the finite-dimensional example are presented through multiple figures with color-coded markers. In the printed text, the colors are referenced (e.g., 'red dots,' 'blue dots'), but a reader without access to the color figures may find it difficult to follow the description. Consider referencing markers by shape or adding a short legend in each figure.
[Section 6.1, first case] The text says 'two fold points in the value of κint are encountered on the way to 0 in the final continuation run' but does not explain how the continuation handles these folds. Since the paper's main algorithm relies on the ability to drive κ to zero, a brief remark about the handling of folds would be useful.
[General] The paper frequently refers to reference [18] for implementation details of adjoint construction and for previous observations about non-monotone continuation. Since [18] is the authors' own work, the dependence is understandable, but the manuscript should make clear which specific claims rely on [18] and which are new in this paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the continuation construction is self-contained, the finite-dimensional results are checked against independent analytical solutions, and the cited self-work is only an implementation dependency.

full rationale

The paper's derivation chain is self-contained. The augmented continuation system (4.1) is constructed directly from the KKT conditions (2.2)-(2.4), with the Fischer-Burmeister function used only to rewrite complementarity conditions as equations via the equivalence (2.5)-(2.7); no fitted parameter is later renamed as a prediction. The Appendix A lemmas are local implicit-function and branch-point arguments whose hypotheses (full-rank Fredholm maps, generic transversality) are stated independently of the target stationary point, and they are proved rather than imported from prior work. In Section 3, the optimum u=(5/3,1/3) is first obtained analytically by solving the KKT conditions (3.2), and the continuation results are explicitly reported as 'consistent with the analytical solution,' so the numerical computation is not used to define the target of the derivation. The only author-relevant citation, [18], concerns the COCO staged-construction implementation and adjoint library; it is invoked for software compatibility and discretization details, not to justify the mathematical existence of the continuation stages. Section 7's concession that it was 'tacitly assumed that each successive stage of continuation is able to drive the appropriate continuation parameters to their desired values' identifies a real reachability limitation and correctness risk, but it is not a circular step: the claimed output is not equivalent by construction to an input of the method. No step satisfying the quote-and-reduction standard for circularity was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method's central claim rests on user-chosen algorithmic parameters (initial point, index sets, computational domain, continuation order), generic transversality assumptions in the lemmas, and the unproven reachability of each continuation stage's target parameter values. No new physical entities are introduced.

free parameters (4)

Initial guess u0 = e.g., (3,1), (−4,0), (1,2), (1,−2), (0,0,0.1,0), p=0
The method requires a user-selected initial point; the paper states 'We are not able to propose a systematic selection algorithm beyond the principles outlined above' (Section 4). Different u0 choices lead to different continuation paths, some of which fail (e.g., FP2 branch in Section 6.1).
Index sets I and P = e.g., I={4}, P={1}; I={3,4}, P=∅; various in Section 6
Selection of initially fixed/released continuation parameters determines the dimension of the first manifold and which singularities are encountered; the paper notes 'different choices may yield distinct locally optimal solutions' and that 'the different scenarios may not be identified a priori' (Section 4).
Computational domain bounds = e.g., −0.2≤p1≤3.5, −0.2≤p2≤1.5, −0.2≤p3≤1.0, 0≤J≤1.5 in Section 6.1
Some continuation runs fail to reach target parameter values within the bounds; e.g., 'If we allow larger computational domain, it turns out that we can arrive at νp2=0 by continuing in the other direction' (Section 6.1, FP2 case).
Continuation order = e.g., drive νJ to 1, then νp3, then κint; or νJ, νp2, νp3
The order in which parameters are driven to targets affects whether singular points are hit; the paper says 'We leave the effects of the imposed order of constraints to future studies' (Section 4).

assumptions (5)

domain assumption Constraint qualification (range condition) at the optimum
Section 2 states the range of (DΦ(û), DG_A(û)) equals Y×R^{|A|} to guarantee existence and uniqueness of KKT multipliers via Corollary 9.4 in [1]; this is a standard constraint qualification but is not verified in the examples.
standard math Fischer-Burmeister NCP function properties
Equations (2.5) through (2.9) use the known equivalence χ(a,b)=0 iff a,b≥0, ab=0 and the inversion formula (2.8) for κ>0; these are standard properties of the Fischer-Burmeister function.
ad hoc to paper Transversality and genericity conditions in Lemmas A.1 through A.6
The lemmas assume DFred(u0) is full rank with one-dimensional nullspace (or bijective) and that (DΨ1(u0))* is linearly independent of (D Fred(u0))*; these are genericity hypotheses introduced for the proofs and are not verified for the numerical examples.
domain assumption Reachability of continuation targets
Section 7 concedes 'It has been tacitly assumed that each successive stage of continuation is able to drive the appropriate continuation parameters to their desired values, preferably monotonically. But the examples showed that this may not be possible within a given computational domain.' This is a load-bearing assumption about the numerical path, not guaranteed by the lemmas.
domain assumption Discretization and adjoint consistency
In Sections 5 and 6, infinite-dimensional problems are discretized via collocation, and adjoints are computed via the COCO library from [18]; validity of the discretized adjoints for the inequality constraints is assumed but not analyzed.

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Pith. "Pith review of Optimization with Equality and Inequality Constraints Using Parameter Continuation." pith.science (2026). https://pith.science/paper/OB2HQVXR

@misc{pith2026190901422,
  author       = {Pith},
  title        = {Pith review of: Optimization with Equality and Inequality Constraints Using Parameter Continuation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OB2HQVXR}},
  note         = {Machine review of arXiv:1909.01422}
}

read the original abstract

We generalize the successive continuation paradigm introduced by Kern\'evez and Doedel [16] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the COCO software package. This is evidenced by a modified form of the COCO core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating stationary solutions of an objective function along families of two-point boundary value problems and in optimal control problems.

Figures

Figures reproduced from arXiv: 1909.01422 by the authors.

**Figure 2.** Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate planes. Gray planes are used to represent tight constraints. Starting from u0 “ p3, 1q and holding κ1 and κ2 fixed at 8 and 0, respectively, a fold poin… view at source ↗

**Figure 3.** Illustration of the pseudo-arclength algorithm for one-dimensional continuation. Starting from a point i located on a solution manifold, the next point i ` 1 is obtained by two steps in the algorithm. In the first step, a predictor denoted by a red cross is generated along the tangent direction at point i. A projection condition is then applied in the second step to locate the point i`1 on the manifold. Singular poi… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate pl…

**Figure 5.** Figure 5: Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate pl…

**Figure 6.** Figure 6: Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate pl…

**Figure 7.** Figure 7: Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate pl…

**Figure 8.** Figure 8: Projections of continuation paths associated with a successive search for stationary solutions. Here, dark green thin lines and hollow markers are used to denote projections of black thick lines and filled markers in three-dimensional space onto the three coordinate pl…

**Figure 9.** Figure 9: Schematic of dynamical system corresponding to (6.6) (adapted from [19]). In order to explore the effects of boundedness of input and output, we first solve the optimal control problem in the absence of such bounds. It follows that q “ 0 and P “ H. We set p1,0 “ ¨ ¨ ¨ …

**Figure 10.** Figure 10: Optimal time histories for θptq (left panel) and xptq (right panel) in the case without inequality constraints (solid lines), with input integral inequality (dashed lines), and with output integral inequality (dotted lines). 0 0.5 1 1.5 2 t -5 0 5 10 15 u(t) No Bound …

**Figure 11.** Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ben-Tal and J

A. Ben-Tal and J. Zowe, A uniﬁed theory of ﬁrst and second order conditions for extremum problems in topological vector spaces, in Optimality and Stability in Mathematical Programming, Springer, 1982, pp. 39–76, https://doi.org/10.1007/BFb0120982

doi:10.1007/bfb0120982 1982

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