Learned Preconditioning for a Primal-Dual Interior-Point Method

Abhinav Madabhushi; Jialin Liu; Minxin Zhang

math.OCSep 28, 2026

Learned Preconditioning for a Primal-Dual Interior-Point Method

Authors: Abhinav Madabhushi, Jialin Liu, Minxin Zhang

Organizations: Department of Mathematics, University of California, Los Angeles · School of Data, Mathematical, and Statistical Sciences, University of Central Florida

Abstract

Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic barriers near constraint boundaries makes IPMs highly sensitive to perturbations, complicating both warm starting and learning reliable updates. We introduce pdLIP, an IPM for smooth nonlinear programs that integrates learned preconditioning with pdProj, an all-shifted primal-dual projected-search IPM. A shared coordinate-wise recurrent network predicts a positive diagonal preconditioner that scales the right-hand side of the reduced Newton system for the primal step, and the remaining slack and multiplier directions are recovered analytically. The learned iterations avoid Hessian evaluations and Newton-system solves, using only first-order and coordinate-wise operations amenable to GPU parallelization. Training is self-supervised, with a loss based on a penalty-barrier merit function and the residual of perturbed optimality conditions, requiring neither target directions nor precomputed solutions. Primal and dual shifts mitigate the barrier's sensitivity to perturbations near constraint boundaries, enabling effective warm starting. Across four classes of 200-dimensional convex and nonconvex constrained problems, pdLIP warm starts reduce pdProj refinement iterations by 63-67% compared with cold starts at the same KKT residual tolerance of $10^{-8}$ , with negligible warm-start generation cost relative to the subsequent pdProj solve. Improvements persist on box-constrained QPs with 1000 variables and extend to applications including portfolio optimization, support vector machines, and a nonlinear control example.

Figures & tables

Figure 1: Overview of one pdLIP iteration. At iteration $k$ , the shared LSTM–MLP maps each $\phi_{k,j}$ to a raw scaling $\widehat{p}_{k,j}$ . The transformation $p_{k,j}=|\widehat{p}_{k,j}|+\delta$ yields a positive diagonal scaling $P_{k}\succ 0$ , which defines the primal direction $\Delta x_{k}=-P_{k}q_{k}$ . The remaining components of $\Delta v_{k}$ are recovered analytically from the pdProj equations, after which the full trial step is projected onto $\Omega_{k}$ . The estimates and barrier and penalty parameters are then updated based on pdProj $O/M/F$ rules.

Class	IPOPT	Cold pdProj	pdLIP + pdProj			Reduction (Iter. / Time)
	Iter.	Iter. / Time	WS Cost	Iter. / Time	Total Time	pdLIP	IPM-LSTM
C-RHS	$15.63$	$13.41/7.53$ s	$20.11$ ms	$4.64/2.58$ s	$2.60$ s	$\mathbf{65.40/65.48}\%$	$42.4/15.1\%$
C-ALL	$15.82$	$13.22/7.41$ s	$19.93$ ms	$4.35/2.42$ s	$2.44$ s	$\mathbf{67.10/67.11}\%$	$35.7/7.5\%$
NC-RHS	$15.60$	$13.47/7.53$ s	$20.03$ ms	$4.93/2.78$ s	$2.80$ s	$\mathbf{63.38/62.81}\%$	$27.5/1.3\%$
NC-ALL	$15.72$	$13.46/8.85$ s	$20.20$ ms	$4.84/2.79$ s	$2.81$ s	$\mathbf{64.04/68.29}\%$	$15.4/-8.6\%$

Table 1: Warm-start results on the constrained benchmark problems from ( Gao et al., 2024 ) .

Dim.	IPOPT	Cold pdProj	pdLIP + pdProj			Reduction
	Iter.	Iter. / Time	WS Cost	Iter. / Time	Total Time	Iter. / Time
$n=10$	$8.99$	$7.45/6.66$ ms	$0.98$ ms	$1.52/1.80$ ms	$2.78$ ms	$\mathbf{79.55/58.21}\%$
$n=200$	$13.48$	$11.44/6.62$ s	$7.32$ ms	$2.19/1.17$ s	$1.18$ s	$\mathbf{80.81/82.17}\%$
$n=1000$	$15.50$	$13.87/13.74$ s	$40.94$ ms	$7.23/4.53$ s	$4.57$ s	$\mathbf{47.90/66.73}\%$

Table 2: Warm-start results on convex box-constrained QPs across dimensions.

Problem	$(s,t)$	IPOPT	Cold pdProj	pdLIP + pdProj			Reduction
		Iter.	Iter. / Time	WS Cost	Iter. / Time	Total Time	Iter. / Time
Portfolio	$(50,5)$	$14.68$	$8.99/22.94$ ms	$5.16$ ms	$3.48/7.82$ ms	$12.98$ ms	$\mathbf{61.28/43.43}\%$
Portfolio	$(200,20)$	$21.06$	$12.07/6.50$ s	$14.07$ ms	$9.36/5.14$ s	$5.15$ s	$\mathbf{22.40/20.71}\%$
SVM	$(5,50)$	$12.52$	$9.30/3.45$ s	$11.57$ ms	$4.52/1.66$ s	$1.67$ s	$\mathbf{51.39/51.55}\%$
SVM	$(20,200)$	$15.46$	$13.83/10.21$ s	$35.95$ ms	$6.31/3.88$ s	$3.92$ s	$\mathbf{54.42/61.65}\%$

Table 3: Warm-start results on portfolio optimization and SVM problems.

Problem	IPOPT	Cold pdProj	pdLIP + pdProj			Reduction
	Iter.	Iter. / Time	WS Cost	Iter. / Time	Total Time	Iter. / Time
Quadrotor	$9.20$	$9.74/19.05$ s	$71.99$ ms	$8.00/15.74$ s	$15.81$ s	$\mathbf{17.89/17.00}\%$

Table 4: Warm-start results on the

200

-dimensional quadrotor optimization problem.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$10^{-4}$	$10^{4}$	$0.0071$	$0.0027$	$0.0076$
$2$	$0.1$	$10^{3}$	$0.0013$	$0.0090$	$0.0091$
$3$	$0.1$	$5\times 10^{3}$	$0.0134$	$0.0024$	$0.0136$
$4$	$10^{-3}$	$10^{3}$	$0.0092$	$0.0133$	$0.0162$
$5$	$0.01$	$10^{3}$	$0.0074$	$0.0211$	$0.0223$

Appendix

Table 5: Hyperparameter sensitivity for convex QP RHS,

n=50,\;m_{\mathrm{eq}}=25,\;m_{\mathrm{ineq}}=25

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.01$	$5\times 10^{3}$	$0.0013$	$0.0018$	$0.0022$
$2$	$10^{-3}$	$5\times 10^{3}$	$0.0328$	$0.0034$	$0.0330$
$3$	$10^{-3}$	$10^{3}$	$0.0335$	$0.0373$	$0.0501$
$4$	$0.1$	$5\times 10^{3}$	$0.0487$	$0.0142$	$0.0507$
$5$	$0.01$	$10^{3}$	$0.1082$	$0.0278$	$0.1117$

Appendix

Table 6: Hyperparameter sensitivity for convex QP ALL,

n=50,\;m_{\mathrm{eq}}=25,\;m_{\mathrm{ineq}}=25

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.1$	$5\times 10^{3}$	$0.0072$	$0.0030$	$0.0078$
$2$	$0.1$	$10^{3}$	$0.0014$	$0.0086$	$0.0087$
$3$	$10^{-3}$	$10^{3}$	$0.0083$	$0.0091$	$0.0123$
$4$	$0.01$	$5\times 10^{3}$	$0.0326$	$0.0035$	$0.0327$
$5$	$0.01$	$10^{4}$	$0.0507$	$0.0065$	$0.0511$

Appendix

Table 7: Hyperparameter sensitivity for nonconvex QP RHS,

n=50,\;m_{\mathrm{eq}}=25,\;m_{\mathrm{ineq}}=25

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$10^{-3}$	$10^{3}$	$0.0062$	$0.0119$	$0.0134$
$2$	$10^{-3}$	$5\times 10^{3}$	$0.0119$	$0.0079$	$0.0143$
$3$	$0.1$	$5\times 10^{3}$	$0.0135$	$0.0074$	$0.0154$
$4$	$0.01$	$5\times 10^{3}$	$0.0154$	$0.0073$	$0.0171$
$5$	$0.1$	$10^{4}$	$0.0183$	$0.0073$	$0.0197$

Appendix

Table 8: Hyperparameter sensitivity for nonconvex QP ALL,

n=50,\;m_{\mathrm{eq}}=25,\;m_{\mathrm{ineq}}=25

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.01$	$10^{4}$	$0.0084$	$0.0061$	$0.0103$
$2$	$10^{-3}$	$10^{4}$	$0.0112$	$0.0062$	$0.0128$
$3$	$0.01$	$5\times 10^{3}$	$0.0101$	$0.0101$	$0.0142$
$4$	$10^{-3}$	$10^{3}$	$0.0198$	$0.0368$	$0.0418$
$5$	$0.1$	$10^{4}$	$0.0471$	$0.0068$	$0.0476$

Appendix

Table 9: Hyperparameter sensitivity for convex QP RHS,

n=200,\;m_{\mathrm{eq}}=100,\;m_{\mathrm{ineq}}=100

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$10^{-3}$	$10^{4}$	$0.0057$	$0.0057$	$0.0080$
$2$	$0.01$	$10^{4}$	$0.0069$	$0.0059$	$0.0091$
$3$	$10^{-3}$	$5\times 10^{3}$	$0.0042$	$0.0096$	$0.0105$
$4$	$0.01$	$5\times 10^{3}$	$0.0208$	$0.0106$	$0.0233$
$5$	$0.01$	$10^{3}$	$0.0053$	$0.0350$	$0.0354$

Appendix

Table 10: Hyperparameter sensitivity for convex QP ALL,

n=200,\;m_{\mathrm{eq}}=100,\;m_{\mathrm{ineq}}=100

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.01$	$5\times 10^{3}$	$0.0211$	$0.0152$	$0.0260$
$2$	$0.1$	$5\times 10^{3}$	$0.0203$	$0.0166$	$0.0262$
$3$	$0.1$	$10^{4}$	$0.0340$	$0.0110$	$0.0358$
$4$	$10^{-3}$	$5\times 10^{3}$	$0.0502$	$0.0186$	$0.0535$
$5$	$10^{-3}$	$10^{3}$	$0.0205$	$0.0512$	$0.0551$

Appendix

Table 11: Hyperparameter sensitivity for nonconvex QP RHS,

n=200,\;m_{\mathrm{eq}}=100,\;m_{\mathrm{ineq}}=100

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.01$	$5\times 10^{3}$	$0.0214$	$0.0236$	$0.0318$
$2$	$10^{-3}$	$5\times 10^{3}$	$0.0303$	$0.0220$	$0.0375$
$3$	$0.01$	$10^{4}$	$0.0400$	$0.0138$	$0.0423$
$4$	$0.1$	$10^{4}$	$0.0408$	$0.0152$	$0.0436$
$5$	$10^{-3}$	$10^{3}$	$0.0152$	$0.0508$	$0.0530$

Appendix

Table 12: Hyperparameter sensitivity for nonconvex QP ALL,

n=200,\;m_{\mathrm{eq}}=100,\;m_{\mathrm{ineq}}=100

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	$\boldsymbol{\tau_{M}}$	LR	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$0.01$	$0.5$	$1$	$10^{-4}$	$0.0019$	$0.0092$	$0.0094$
$2$	$0.01$	$2$	$1$	$10^{-4}$	$0.0035$	$0.0110$	$0.0116$
$3$	$0.01$	$1$	$1$	$10^{-4}$	$0.0038$	$0.0120$	$0.0126$
$4$	$0.01$	$2$	$5$	$10^{-4}$	$0.0039$	$0.0134$	$0.0139$
$5$	$0.01$	$1$	$5$	$10^{-4}$	$0.0092$	$0.0225$	$0.0243$

Appendix

Table 13: Hyperparameter sensitivity for portfolio problems,

s=50,\;t=5

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	$\boldsymbol{\tau_{M}}$	LR	Mean $e_{P}$	Mean $e_{D}$	Combined
$1$	$0.01$	$0.1$	$1$	$10^{-4}$	$0.0091$	$0.0988$	$0.0992$
$2$	$0.01$	$0.5$	$1$	$10^{-4}$	$0.0101$	$0.1170$	$0.1175$
$3$	$10^{-3}$	$5\times 10^{3}$	$1$	$10^{-4}$	$0.0569$	$0.1688$	$0.1782$
$4$	$0.01$	$2$	$1$	$10^{-4}$	$0.0135$	$0.1800$	$0.1805$
$5$	$0.01$	$1$	$1$	$10^{-4}$	$0.0074$	$0.2256$	$0.2257$

Appendix

Table 14: Hyperparameter sensitivity for portfolio problems,

s=200,\;t=20

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$10^{-3}$	$0.5$	$0.0003$	$0.0086$	$0.0086$
$2$	$0.1$	$0.5$	$0.0005$	$0.0200$	$0.0200$
$3$	$0.01$	$2$	$0.0005$	$0.0312$	$0.0312$
$4$	$0.1$	$2$	$0.0006$	$0.0761$	$0.0761$
$5$	$10^{-3}$	$2$	$0.0023$	$0.0969$	$0.0969$

Appendix

Table 15: Hyperparameter sensitivity for SVM problems,

s=5,\;t=50

Rank	$\boldsymbol{\mu_{B}^{0}}$	$\boldsymbol{\mu_{P}^{0}}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
$1$	$10^{-3}$	$0.5$	$0.0002$	$0.0324$	$0.0324$
$2$	$0.01$	$0.5$	$0.0007$	$0.1345$	$0.1345$
$3$	$10^{-3}$	$5$	$0.0002$	$0.1726$	$0.1726$
$4$	$0.1$	$2$	$0.0039$	$0.1805$	$0.1805$
$5$	$10^{-3}$	$2$	$0.0065$	$0.3435$	$0.3436$

Appendix

Table 16: Hyperparameter sensitivity for SVM problems,

s=20,\;t=200

Rank	$\mu_{B}^{0}$	$\mu_{P}^{0}$	Mean $e_{P}$	Mean $e_{D}$	$e_{\mathrm{comb}}$
1	$10^{-2}$	$0.1$	$4.0483\times 10^{-5}$	$0.0304$	$0.0304$
2	$10^{-2}$	$2$	$0.0096$	$0.0414$	$0.0426$
3	$10^{-3}$	$2$	$0.0082$	$0.0460$	$0.0467$
4	$10^{-3}$	$10$	$0.0299$	$0.0811$	$0.0865$
5	$10^{-2}$	$10$	$0.0352$	$0.1168$	$0.1220$

Appendix

Table 17: Hyperparameter sensitivity for single-quadrotor navigation,

n=200

and

T=11

Problem Class	Method	Iter.	Grad. Calls	Func. Calls
Convex QP RHS	IPOPT	$15.63$	$17.63$	$16.63$
	Cold pdProj	$13.41$	$16.73$	$16.73$
	Warm-started pdProj	$4.64$	$5.64$	$5.64$
Convex QP ALL	IPOPT	$15.82$	$17.82$	$16.82$
	Cold pdProj	$13.22$	$17.65$	$17.65$
	Warm-started pdProj	$4.35$	$5.36$	$5.36$

Appendix

Table 18: Iterations, gradient calls, and function calls for the

n=200

constrained benchmarks.

Problem Class	Method	Iter.	Grad. Calls	Func. Calls
Convex QP RHS	IPOPT	$12.08$	$14.08$	$13.08$
	Cold pdProj	$8.97$	$10.38$	$10.38$
	Warm-started pdProj	$2.92$	$3.93$	$3.93$
	Iteration reduction	$\mathbf{67.49\%}$	–	–
Convex QP ALL	IPOPT	$12.17$	$14.17$	$13.17$
	Cold pdProj	$9.31$	$11.00$	$11.00$

Appendix

Table 19: Iterations, gradient calls, and function calls for the

n=50

constrained benchmarks.

Problem Class	Cold Time	WS Cost	Total Time	Time Red.
Convex QP RHS	$2.69$ s	$5.23$ ms	$878$ ms	$\mathbf{67.31\%}$
Convex QP ALL	$2.80$ s	$4.59$ ms	$861$ ms	$\mathbf{69.22\%}$
Nonconvex RHS	$2.81$ s	$5.31$ ms	$886$ ms	$\mathbf{68.44\%}$
Nonconvex ALL	$2.84$ s	$5.64$ ms	$917$ ms	$\mathbf{67.75\%}$

Appendix

Table 20: Runtime results for the

n=50

constrained benchmarks.

Problem	Method	Iter.	Grad. Calls	Func. Calls
$n=200$	IPOPT	$13.48$	$15.48$	$14.48$
	Cold pdProj	$11.44$	$12.44$	$12.44$
	Warm-started pdProj	$2.19$	$3.19$	$3.19$
$n=1000$	IPOPT	$15.50$	$17.50$	$16.50$
	Cold pdProj	$13.87$	$14.87$	$14.87$
	Warm-started pdProj	$7.23$	$8.23$	$8.23$

Appendix

Table 21: Iterations, gradient calls, and function calls for the box-constrained QPs.

Problem	$(s,t)$	Method	Iter.	Grad. Calls	Func. Calls
Portfolio	$(50,5)$	IPOPT	$14.68$	$16.68$	$15.68$
		Cold pdProj	$8.99$	$9.99$	$9.99$
		Warm pdProj	$3.48$	$4.48$	$4.48$
Portfolio	$(200,20)$	IPOPT	$21.06$	$23.06$	$22.06$
		Cold pdProj	$12.07$	$13.18$	$13.18$
		Warm pdProj	$9.36$	$10.57$	$10.57$

Appendix

Table 22: Iterations, gradient calls, and function calls for the portfolio and SVM benchmarks.

Problem	Method	Iter.	Grad. Calls	Func. Calls
Quadrotor	IPOPT	$9.20$	$11.20$	$10.20$
	Cold pdProj	$9.74$	$10.74$	$10.74$
	Warm-started pdProj	$8.00$	$9.00$	$9.00$

Appendix

Table 23: Iterations, gradient calls, and function calls for the nonlinear quadrotor control problem.

Problem	Method	Converged	Avg. Runtime
Box QP, $n=200$	pdLIP	$100\%$	$4.02$ ms
	IPOPT	$100\%$	$85.27$ ms
Constrained QP RHS, $n=200$	pdLIP	$72.4\%$	$20.11$ ms
	IPOPT	$100\%$	$315.54$ ms
Constrained QP ALL, $n=200$	pdLIP	$85.2\%$	$19.93$ ms
	IPOPT	$100\%$	$344.84$ ms

Appendix

Table 24: Direct approximate-solution results on the larger QP benchmarks.

Problem	Method	Converged	Avg. Runtime
Convex RHS	pdLIP	$95.3\%$	$5.23$ ms
	IPOPT	$100\%$	$4.29$ ms
Convex ALL	pdLIP	$91.7\%$	$4.59$ ms
	IPOPT	$100\%$	$4.43$ ms
Nonconvex RHS	pdLIP	$92.5\%$	$5.31$ ms
	IPOPT	$100\%$	$4.37$ ms

Appendix

Table 25: Direct approximate-solution results on the

n=50

constrained benchmarks.

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NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guarantees

Bimol Nath Roy, Rahul Golder, MM Faruque Hasan

Texas A&M University

Inequality Constraints Neural Network Optimization Feasibility Learn

Open paper AskPDF arXiv

Nonlinear Parametric Optimization Network (NLPOpt-Net) is an unsupervised learning architecture to solve constrained nonlinear programs (NLP). Given the structure of an NLP, it learns the parametric solution maps with guaranteed constraint satisfaction. The architecture consists of a backbone neural network (NN) followed by a multilayer (

k

-layered) projection. While the NN drives toward optimality through a loss function consisting of a modified Lagrangian augmented with a consistency loss, the projection ensures feasibility by projecting the NN predictions in the original constraint manifold. Instead of typical distance minimization, our projection exploits local quadratic approximations of the original NLP. Under certain conditions (such as convexity), the projection has a descent property, which improves the NN predictions further. NLPOpt-Net deploys an inversion-free, modified Chambolle-Pock algorithm to solve the constrained quadratic projections during the forward pass and uses the implicit function theorem for efficient backpropagation. The fixed structure of the projection further allows decoupling of the NN and the projection once the training is complete. NLPOpt-Net solves large-scale convex QP, QCQP, NLP, and nonconvex problems with near zero optimality gap and constraint violations reduced to machine precision. Additionally, it provides near accurate prediction of the active sets and corresponding dual variables, thereby enabling a scalable approach for multiparametric programming. Compiling the projection in C provides order of magnitude improvement in inference time compared to JAX. We provide the codes and NLPOpt-Net as a ready to use package that includes GPU support.

Abstract

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