Scen-Opt: A Scenario Optimization Toolbox for Data-Driven Convex Programming

Ben Wooding; Simone Garatti; Marco C. Campi; Abolfazl Lavaei

cs.MSOct 6, 2026

Scen-Opt: A Scenario Optimization Toolbox for Data-Driven Convex Programming

Authors: Ben Wooding, Simone Garatti, Marco C. Campi, Abolfazl Lavaei

Organizations: Institute for Convergent Software Integrated Systems, Vanderbilt University, USA · Department of Electronics, Information and Bioengineering, Politecnico di Milano, Italy · Department of Information Engineering, University of Brescia, Italy · School of Computing, Newcastle University, UK

Abstract

The scenario approach is a well-established statistical framework for data-driven decision-making. In particular, in data-driven optimization, the scenario approach unveils how the problem structure governs out-of-sample generalization, and offers a principled basis for assessing and certifying the reliability of the optimal solution as per constraint satisfaction. Despite its strong theoretical development and wide applicability, no software toolbox has been available to date that enables user-friendly, data-driven convex optimization within the scenario-approach framework. In this paper, we introduce Scen-Opt, an open-source software tool that integrates convex programming with data samples while providing statistical guarantees grounded in scenario theory. Scen-Opt is implemented in Python, supporting data-driven linear, quadratic, and semidefinite programming, and offers a Python-based web application with an intuitive and reactive graphical user interface (GUI) built using modern web technologies. Scen-Opt can be used directly through its online interface or installed locally, accommodating both manual input and data-file uploads (CSV, JSON, TXT, TSV, MAT, Excel, NPY, NPZ, Parquet). Built on a Python backend with a modern JavaScript frontend, Scen-Opt offers a highly user-friendly experience and efficient usability across desktops, laptops, tablets, and mobile devices. In this paper, Scen-Opt is applied to a set of representative benchmarks, demonstrating its practical effectiveness for data-driven convex optimization with guaranteed performance.

Figures & tables

Panel of Figure 1 — Figure 1 : Robust LP: Smallest enclosing interval example.

Figure 2 : Robust QP: Support vector machine example.

Figure 3 : Robust SDP: LPV System Example.

Figure 4 : Web interface of Scen-Opt illustrating the relaxed and regularized LP.

Figure 5 : Scen-Opt web interface displaying the results table for the relaxed and regularized LP set up in Fig. 4 ( $d=1$ , $N=9$ scenarios, $\beta=0.1$ ), with one column for each pair $(\rho,\tau)$ , $\rho,\tau\in\{0.2,0.3\}$ .

Panel of Figure 6 — Figure 6 : Example graphical output from Scen-Opt for cases involving relaxation or regularization, for the run of Figs. 4 and 5 .

Benchmark	$d$	$q$	$m_{s}$	$m_{h}$	$\rho$	$\tau$	$N$	$k$	$\underline{\varepsilon}$	$\bar{\varepsilon}$	Time (s)
Linear Programs (LP)
Half Width	2	1	2	0	$0$	$0$	100	2	$0$	$0.2085$	1.2
Growth Bound	12	6	3	6	$0$	$0$	3127	6	N/A	$0.0103$	1276.8
Inventory	11	15	6	17	$100^{\dagger}$	$0$	500	7	$0$	$0.0666$	4.0
Portfolio CVaR	13	12	4	27	$0.016^{\dagger}$	$0$	1255	3	$0$	$0.0203$	22.0
Power Dispatch	120	48	48	330	$100^{\dagger}$	$0$	150	34	$0.0791$	$0.4496$	2.9

Table 1 : Summary of all twelve benchmark case studies solved with Scen-Opt , grouped by program type.

d

= decision variables,

q

= uncertainty dimension,

m_{s}

= scenario constraints (rows of

A(\delta)

for LP/QP, LMI dimension for SDP),

m_{h}

= hard constraints (rows of

G

for LP/QP, LMI dimension for SDP),

N

= number of sampled scenarios,

k

= complexity (number of support constraints). All benchmarks use the confidence parameter

\beta=10^{-6}

. Solve times are wall-clock seconds on a desktop with an Intel Core Ultra 9 285K (24 cores) and 64 GB RAM running Windows 11. † Power Dispatch, Inventory and Portfolio CVaR use an augmented formulation in which the slack penalty is embedded in the objective coefficient

c

; the value of rho passed to the solver is therefore

0

, while the displayed value reflects the user-facing penalty weight. N/A = degeneracy detected, so the lower risk bound is not certified. Scenario data come from Yahoo Finance (Portfolio CVaR) and the UCI repository (Iris SVM Fisher (1936) , Covariance Estimation Aeberhard and Forina (1992) , Min. Encl. Ellipsoid Wolberg et al. (1993) ); Radiation Therapy uses synthetic data inspired by TROTS Breedveld and Heijmen (2019) , and the scenarios of all other benchmarks are generated as described in Appendix A .

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7 : Left. Comparison of optimal order against mean demand. Center. Quantified risk bounds for $N=500$ and confidence $99\%$ . Right. Graphic of scenario samples for different produce.

Product	Cost ($)	Mean Demand (cases)	Yield ( $\%$ )	Space (cu. ft)
Strawberries	18.50	85	75 (25 loss)	1.2
Tomatoes	14.25	120	88 (12 loss)	1.5
Lettuce	12.00	95	82 (18 loss)	2.0
Avocados	32.00	60	90 (10 loss)	0.8
Bell Peppers	22.50	75	85 (15 loss)	1.3

Appendix

Table 2 : Produce purchasing details. The data is synthetic but calibrated to real data Buzby et al. (2014) ; Perez et al. (2017) , along with USDA Terminal Market Prices for highest grade products.

Product	Order (cases)	Mean Demand	Service Level ( $\%$ )
Strawberries	150	83.9	85.8
Tomatoes	155.2	118.9	72.4
Lettuce	40.4	94.3	0.2
Avocados	118.9	59.8	99.8
Bell Peppers	121.0	74.1	90.0

Appendix

Table 3 : Optimal Purchasing Order

Generator	Capacity	Min Output	Cost	Ramp Rate
Gas 1	400 MW	100 MW	40 $/MW	200 MW/h
Gas 2	300 MW	75 MW	50 $/MW	150 MW/h
Coal	500 MW	200 MW	30 $/MW	100 MW/h

Appendix

Table 4 : Thermal generator parameters. Costs and capacities are synthetic but calibrated against standard values.

Figure 8 : Left. Stacked generation dispatch over $24$ hours: coal provides baseload, gas ramps for the evening peak. Center. Quantified risk bounds for $N=150$ and confidence $99\%$ . Right. Wind and solar scenario variability (median with $25-75$ th and $10-90$ th percentile bands).

Ticker	Asset	Class	Role
SPY	S&P 500 ETF	US Equity	Core Growth
AGG	US Aggregate Bond	Fixed Income	Stability
VNQ	Real Estate REIT	Real Assets	Inflation Hedge
GLD	Gold	Commodities	Crisis Hedge
EFA	Intl Developed	Intl Equity	Diversification
TLT	20+ Year Treasury	Long Bonds	Duration

Appendix

Table 5 : ETF asset universe. Historical returns are downloaded from Yahoo Finance via yfinance (5 years daily data).

Figure 9 : Left. Optimal portfolio weights with position limit constraints. Center. Quantified risk bounds for $N=1255$ and $\beta=10^{-6}$ . Right. Portfolio return distribution with VaR $(95\%)$ and CVaR $(95\%)$ marked.

Figure 10 : Left. Optimal half-width given sampled points. Center. Quantified risk bounds for $N=100$ and confidence $99.9999\%$ . Right. Density of sampled points across the region.

Figure 11 : Left. Dose-volume histogram showing tumor coverage above the 137.75 Gy minimum with OAR doses within limits (rectum

<100

Gy, bladder

<120

Gy); the

x

-axis is capped at

3\times

prescription as brachytherapy produces extreme hot spots near dwell positions. Center. Risk bounds with

k=5

support constraints. Right. Dwell-position intensity profile showing most channels near maximum with selective suppression near OARs.

Figure 12 : Left. Optimal trajectory arcing above the wall obstacle. Center. Risk bounds as a function of complexity $k=$ . Right. Wall clearance at timesteps within the wall’s $x$ -range.

Figure 13 : Left. Optimal separating hyperplane with margin boundaries and support vectors. Center. Risk bounds with complexity $k=2$ . Right. Petal feature distributions by class.

Figure 14 : Left. Entry-wise error $\hat{\Sigma}-\Sigma_{\mathrm{full}}$ ; diagonal inflation reflects the domination requirement. Center. Risk bounds with $k=7$ support constraints. Right. Eigenvalue spectrum comparison: the estimated covariance uniformly dominates the full-sample covariance in every principal direction.

Figure 15 : Left. 2D projection showing 569/569 points contained (100%) inside the region. Center. Risk bounds with

k=5

support constraints. Right. Containment profile: sorted

(z-\bar{x})^{\top}P\,(z-\bar{x})

values for all 569 data points; the boundary at 1.0 separates interior from exterior points.

Figure 16 : Left. Stabilizing trajectories with for different scenarios $\delta$ . Right. Risk bounds for the LPV stability with complexity $k=1$ .

Figure 17 : Left. Stabilizing trajectories for example scenarios $\delta$ . Center. Risk bounds for the quadratic stability with complexity $k=1$ . Right. Stability margins for each sampled scenario, with red less stable and blue more stable.

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Disciplined Biconvex Programming

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We introduce disciplined biconvex programming (DBCP), a modeling framework for specifying and solving biconvex optimization problems. Biconvex optimization problems arise in various applications, including machine learning, signal processing, computational science, and control. Solving a biconvex optimization problem in practice usually involves heuristic methods based on alternate convex search (ACS), which iteratively optimizes over one block of variables while keeping the other fixed, so that the resulting subproblems are convex and can be efficiently solved. However, designing and implementing an ACS solver for a specific biconvex optimization problem usually requires significant effort from the user, which can be tedious and error-prone. DBCP extends the principles of disciplined convex programming to biconvex problems, allowing users to specify biconvex optimization problems in a natural way based on a small number of syntax rules. The resulting problem can then be automatically split and transformed into convex subproblems, for which a customized ACS solver is then generated and applied. DBCP allows users to quickly experiment with different biconvex problem formulations, without expertise in convex optimization. We implement DBCP in the open-source Python package dbcp, as an extension to the well known domain specific language CVXPY for convex optimization.

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